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SE  7  (Rev.  7/82)                                           UCSD  Libr. 

s^e 


ANALYTIC  GEOMETRY 
OF  SPACE 


BY 


VIRGIL  SNYDER,  Ph.D.  (Gottingen) 

Professor  of  Mathematics  at  Cornell 
University 


C.  H.  SISAM,  Ph.D.  (Cornell) 

Assistant  Professor  of  Mathematics  at  the 
University  of  Ilunois 


NEW  YORK 
HENRY  HOLT  AND   COMPANY 


Copyright,  1914, 

BY 

HENRY  HOLT  AND  COMPANY 
April,  1924 


PRINTED  IN   U.  8.  A. 


PREFACE 

In  this  book,  which  is  planned  for  an  introductory  course, 
the  first  eight  chapters  include  the  subjects  usually  treated  in 
rectangular  coordinates.  They  presuppose  as  much  knowledge 
of  algebra,  geometry,  and  trigonometry  as  is  contained  in  the 
major  requirement  of  the  College  Entrance  Examination  Board, 
and  as  much  plane  analytic  geometry  as  is  contained  in  the 
better  elementary  textbooks.  In  this  portion,  proofs  of  theorems 
from  more  advanced  subjects  in  algebra  are  supplied  as  needed. 
Among  the  features  of  this  part  are  the  development  of  linear 
systems  of  planes,  plane  coordinates,  the  concept  of  infinity,  the 
treatment  of  imaginaries,  and  the  distinction  between  centers 
and  vertices  of  quadric  surfaces.  The  study  of  this  portion  can 
be  regarded  as  a  first  course,  not  demanding  more  than  thirty  or 
forty  lessons. 

In  Chapter  IX  tetrahedral  coordinates  are  introduced  by  means 
of  linear  transformations,  under  which  various  invariant  proper- 
ties are  established.  These  coordinates  are  used  throughout  the 
next  three  chapters.  The  notation  is  so  chosen  that  no  ambigu- 
ity can  arise  between  tetrahedral  and  rectangular  systems.  The 
selection  of  subject  matter  is  such  as  to  be  of  greatest  service  for 
further  study  of  algebraic  geometry. 

In  Chapter  XIII  a  more  advanced  knowledge  of  plane  analytic 
geometry  is  presupposed,  but  the  part  involving  Pliicker's  num- 
bers may  be  omitted  without  disturbing  the  continuity  of  the 
subject.  In  the  last  chapter  extensive  use  is  made  of  the  cal- 
culus, including  the  use  of  partial  differentiation  and  of  the 
element  of  arc. 

The  second  part  will  require  about  fifty  lessons. 


CONTENTS 

CHAPTER   I 
COORDINATES 

ARTICLE  PAGE 

1.  Coordinates 1 

2.  Orthogonal  projection 3 

3.  Direction  cosines  of  a  line 5 

4.  Distance  between  two  points    ........  6 

5.  Angle  between  two  directed  lines     .......  7 

6.  Point  dividing  a  segment  in  a  given  ratio 8 

7.  Polar  coordinates 10 

8.  Cylindrical  coordinates 10 

9.  Spherical  coordinates 11 


CHAPTER   II 

PLANES   AND    LINES 

10.  Equation  of  a  plane 12 

11.  Plane  through  three  points        ........  13 

12.  Intercept  form  of  the  equation  of  a  plane         .....  13 

13.  Normal  form  of  the  equation  of  a  plane .14 

14.  Reduction  of  a  linear  equation  to  the  normal  form  ....  15 

15.  Angle  between  two  planes        ........  16 

16.  Distance  to  a  point  from  a  plane 17 

17.  Equations  of  a  line   ..........  19 

18.  Direction  cosines  of  the  line  of  intersection  of  two  planes       .         .  19 

19.  Forms  of  the  equations  of  a  line 20 

20.  Parametric  equations  of  a  line 21 

21.  Angle  which  a  line  makes  with  a  plane    ......  22 

22.  Distance  from  a  point  to  a  line 23 

23.  Distance  between  two  non-intersec\'ing  lines 24 

24.  System  of  planes  through  a  line 25 

25.  Application  to  descriptive  geometry 28 

V 


VI 


CONTENTS 


ARTirLE  PAGE 

26.  Bundles  of  planes 29 

27.  Plane  coordinates 31 

28.  Equation  of  a  point 32 

29.  Homogeneous  coordinate  of  the  point  and  of  the  plane    ...  33 

30.  Equation  of  the  plane  and  of  the  point  in  homogeneous  coordinates  34 

31.  Equation  of  the  origin.     Coordinates  of  planes  through  the  origin  .  34 

32.  Plane  at  infinity 36 

33.  Lines  at  infinity        ..........  35 

34.  Coordinate  tetrahedron 35 

35.  System  of  four  planes 36 

CHAPTER   III 

t 

TRANSFORMATION  OF  COORDINATES 

36.  Translation 38 

37.  Rotation 38 

38.  Rotation  and  reflection  of  axes 41 

39.  Euler's  formulas  for  rotation  of  axes        ......  42 

40.  Degree  unchanged  by  transformation  of  coordinates        ...  42 


CHAPTER   IV 
TYPES    OF    SURFACES 


41.  Imaginary  points,  lines,  and  planes 

42.  Loci  of  equations 

43.  Cylindrical  surfaces 

44.  Projecting  cylinders 

45.  Plane  sections  of  surfaces 

46.  Cones        .... 

47.  Surfaces  of  revolution 


44 

46 
47 
47 
48 
49 
50 


CHAPTER   V 

THE    SPHERE 

48.  The  equation  of  the  sphere 52 

49.  The  absolute 52 

50.  Tangent  plane 65 

51.  Angle  between  two  spheres 65 

52.  Spheres  satisfying  given  conditions 66 

63.    Linear  systems  of  spheres 67 

54.    Stereographic  projection 59 


CONTENTS 


Vll 


CHAPTER   VI 
FORMS   OF   QUADRIC   SURFACES 

ARTICLE  PAGE 

55.  Definition  of  a  quadric 63 

56.  The  ellipsoid 63 

57.  The  hyperboloid  of  one  sheet 65 

58.  The  hyperboloid  of  two  sheets           .......  67 

59.  The  imaginary  ellipsoid    .........  68 

60.  The  elliptic  paraboloid 69 

61.  The  hyperbolic  paraboloid 70 

62.  The  quadric  cones    ..........  71 

63.  The  quadric  cylinders 72 

64.  Summary 72 


CHAPTER   VII 
CLASSIFICATION    OF    QUADRIC    SURFACES 

65.  Intersection  of  a  quadric  and  a  line 

66.  Diametral  planes,  center  .... 

67.  Equation  of  a  quadric  referred  to  its  center 

68.  Principal  planes        ...... 

69.  Reality  of  the  roots  of  the  discriminating  cubic 

70.  Simplification  of  the  equation  of  a  quadric 

71.  Classification  of  quadric  surfaces 

72.  Invariants  under  motion  ..... 

73.  Proof  that  7,  t/,  and  D  are  invariant 

74.  Proof  that  A  is  invariant 

75.  Discussion  of  numerical  equations    . 


74 

75 
77 
78 
79 
80 
€1 
82 
83 
84 


CHAPTER   VIII 

SOME   PROPERTIES   OF   QUADRIC   SURFACES 

76.  Tangent  lines  and  planes           ........  90 

77.  Normal  forms  of  the  equation  of  the  tangent  plane           ...  91 
78  Normal  to  a  quadric 92 

79.  Rectilinear  generators       .         .         .         .         .         .         .         .         .93 

80.  Asymptotic  cone       ..........  95 

81.  Plane  sections  of  quadrics 96 

82.  Circular  sections 98 

83.  Real  circles  on  types  of  quadrics 100 

84.  Confocal  quadrics 104 

85.  Confocal  quadrics  through  a  point.     Elliptic  coordinates         .         .  105 

86.  Confocal  quadrics  tangent  to  a  line  .......  107 

87.  Confocal  quadrics  in  plane  coordinates 108 


Vlll 


CONTENTS 


CHAPTER  IX 
TETRAHEDRAL   COORDINATES 

ARTICLE  PAOR 

88.  Definition  of  tetrahedral  coordinates 109 

89.  Unit  point 110 

90.  Equation  of  a  plane.     Plane  coordinates        .        .        .        .         .111 

91.  Equation  of  a  point 112 

92.  Equations  of  a  line 112 

93.  Duality 11.3 

94.  Parametric  equations  of  a  plane  and  of  a  point      ....  114 

95.  Parametric  equations  of  a  line.    Range  of  points.    Pencils  of  planes  115 

96.  Transformation  of  point  coordinates 117 

97.  Transformation  of  plane  coordinates      .        .        .         .        .        .119 

98.  Projective  transformations 120 

99.  Invariant  points 121 

100.   Cross-ratio 121 


CHAPTER  X 


QUADRIC  SURFACES  IN  TETRAHEDRAL  COORDINATES 


101.  Form  of  equation    ....... 

102.  Tangent  lines  and  planes 

103.  Condition  that  the  tangent  plane  is  indeterminate  . 

104.  The  invariance  of  the  discriminant 

105.  Lines  on  the  quadric  surface 

106.  Equation  of  a  quadric  in  plane  coordinates    . 

107.  Polar  planes 

108.  Harmonic  property  of  conjugate  points 

109.  Locus  of  points  which  lie  on  their  own  polar  planes 

110.  Tangent  cone  ........ 

111.  Conjugate  lines  as  to  a  quadric        .... 

112.  Self-polar  tetrahedron     ...... 

113.  pjquation  referred  to  a  self-polar  tetrahedron 

114.  Law  of  inertia 

115.  Rectilinear  generators.     Reguli      .... 

116.  Hyperbolic  coordinates.      Parametric  equations     . 

117.  Projection  of  a  quadric  upon  a  plane 

118.  Equations  of  the  projection     ..... 

119.  Quadrics  deterniineii  by  three  non-intersecting  lines 

120.  Transversals  of  four  skew  lines       .... 

121.  The  quatlric  cone    ....... 

122.  Projection  of  a  quadric  coue  upon  a  plane 


124 
124 
125 
126 
129 
130 
132 
132 
133 
133 
134 
135 
135 
136 
137 
138 
139 
140 
141 
143 
143 
145 


CONTENTS  . 


IX 


CHAPTER   XI 
LINEAR   SYSTEMS   OF   QUADRICS 

ARTICLE 

123.  Pencil  of  quadrics 

124.  The  \-discrirainant  .... 

125.  Invariant  factors     ..... 

126.  The  characteristic  ..... 

127.  Pencil  of  quadrics  having  a  common  vertex 

128.  Classification  of  pencils  of  quadrics 

129.  Quadrics  having  a  double  plane  in  common 

130.  Quadrics  having  a  line  of  vertices  in  common 

131.  Quadrics  having  a  vertex  in  common 

132.  Quadrics  having  no  vertex  in  common 

133.  Forms  of  pencils  of  quadrics  . 

134.  Line  conjugate  to  a  point 

135.  Equation  of  the  pencil  in  plane  coordinates 

136.  Bundle  of  quadrics  .... 

137.  Representation  of  the  quadrics  of  a  bundle  by 

138.  Singular  quadrics  of  the  bundle 

139.  Intersection  of  the  bundle  by  a  plane 

140.  The  vertex  locus  J  .... 

141.  Polar  theory  in  the  bundle 

142.  Some  special  bundles       .... 

143.  Webs  of  quadrics    ..... 

144.  The  Jacobian  surface  of  a  web 

145.  Correspondence  with  the  planes  of  space 

146.  Web  with  six  basis  points 

147.  Linear  sy.stems  of  rank  r         .         .         . 

148.  Linear  systems  of  rank  r  in  plane  coordinates 

149.  Apolarity        ...... 

150.  Linear  sy.stems  of  apola,r  quadrics  . 


the  points 


of  a 


plane 


PAGE 

147 
147 
148 
150 
161 
161 
161 
151 
152 
156 
163 
165 
166 
167 
168 
168 
169 
170 
171 
173 
175 
175 
177 
177 
180 
181 
181 
186 


CHAPTER   XII 

TRANSFORMATIONS   OF   SPACE 

161.    Projective  metric 188 

152.  Pole  and  polar  as  to  the  absolute    .......  188 

153.  Equations  of  motion 190 

154.  Classification  of  projective  transformations 191 

155.  Standard  forms  of  equations  of  projective  transformations     .        .  196 

156.  Birational  transformations      . 196 

157.  Quadratic  transformations       ........  198 

158.  Quadratic  inversion         .........  201 

159.  Transformation  by  reciprocal  radii 201 

160.  Cyclides 203 


CONTENTS 


CHAPTER   XIII 

CURVES    AND    SURFACES    IN    TETRAHEDRAL 
COORDINATES 

I.     Algebraic  Surfaces 

AKTICLE  PAGE 

161.  Number  of  constants  in  the  equation  of  a  surface  ....  206 

162.  Notation 207 

163.  Intersection  of  a  line  and  a  surface 207 

164.  Polar  surfaces 208 

165.  Tangent  lines  and  planes 209 

166.  Inflexional  tangents 210 

167.  Double  points 210 

168.  The  first  polar  surface  and  tangent  cone 211 

169.  Class  of  a  surface.     Equation  in  plane  coordinates        .        .         .212 

170.  The  Hessian 213 

171.  The  parabolic  curve 214 

172.  The  Steinerian 214 


II.     Algebraic  Space  Curves 

173.  Systems  of  equations  defining  a  space  curve  . 

174.  Order  of  an  algebraic  curve 

175.  Projecting  cones      ....... 

176.  Monoidal  representation  ..... 

177.  Number  of  intersections  of  algebraic  curves  and  surfaces 

178.  Parametric  equations  of  rational  curves 

179.  Tangent  lines  and  developable  surface  of  a  curve  . 

180.  Osculating  planes.     Equation  in  plane  coordinates 

181.  Singular  points,  lines,  and  planes  .... 

182.  The  Cayley-Salmon  formulas  .... 

183.  Curves  on  non-singular  quadric  surfaces 

184.  Space  cubic  curves 

185.  Metric  classification  of  space  cubic  curves 

186.  Classification  of  space  quartic  curves 

187.  Non-singular  quartic  curves  of  the  first  kind  . 

188.  Rational  quartics 


216 

216 
217 
219 
221 
222 
224 
224 
226 
226 
228 
230 
234 
235 
238 
242 


CHAPTER   XIV 

DIFFERENTIAL   GEOMETRY 

I.     Analytic  Curves 

189.  Length  of  arc  of  a  space  curve 245 

190.  The  moving  trihedral 246 

191.  Curvature 248 


CONTENTS 


XI 


ARTICLK 

192.  Torsion    .... 

193.  The  Frenet-Serret  formulas 

194.  The  osculating  sphere 

195.  Minimal  curves 


PAGF. 

249 
250 
251 
252 


II.     Analytic  Surfaces 

196.  Parametric  equations  of  a  surface 

197-  Systems  of  curves  on  a  surface 

198.  Tangent  plane.     Normal  line 

199.  Differential  of  arc  . 

200.  Minimal  curves 

201 .  Angle  between  curves.     Differential  of  surface 

202.  Radius  of  normal  curvature.     Meusnier's  theorem 

203.  Asymptotic  tangents.     Asymptotic  curves 

204.  Conjugate  tangents 

205.  Principal  radii  of  normal  curvature 

206.  Lines  of  curvature  . 

207.  The  indicatrix 

Answers 

Index      .... 


254 
255 
255 
257 
258 
259 
259 
261 
261 
262 
263 
266 

269 
287 


ANALYTIC  GEOMETRY  OF  SPACE 


CHAPTER   I 


COORDINATES 

1.  Rectangular  coordinates.  The  idea  of  rectangular  coordinates 
as  developed  in  plane  analytic  geometry  may  be  extended  to  space 
in  the  following  manner. 

Let  there  be  given  three  mutually  perpendicular  planes 
(Fig.  1)  XOY,  YOZ,  ZOX,  intersecting  at  0,  the  origin.  These 
planes  will  be  called  coordinate  planes.  The  planes  ZOX,  XOY 
intersect  in  X'OX,  the  X-axis;  the  planes  XOY,  YOZ  intersect 
in  Y'OY,  the  F-axis ;  the 
planes  YOZ,  ZOX  intersect 
in  Z'OZ,  the  Z-axis.  Dis- 
tances measured  in  the 
directions  X'OX,  Y'OY, 
Z'OZ,  respectively,  will  be 
considered  positive ;  those 
measured  in  the  opposite 
directions  will  be  regarded 
as  negative.  The  coordi- 
nates of  any  point  P  are  its  distances  from  the  three  coordinate 
planes.  The  distance  from  the  plane  YOZ  is  denoted  by  x,  the 
distance  from  the  plane  ZOX  is  denoted  by  y,  and  the  distance 
from  the  plane  XOY  is  denoted  by  z.  These  three  numbers 
X,  y,  z  are  spoken  of  as  the  x-,  y-,  z-coordinates  of  P,  respect- 
ively. Any  point  P  in  space  has  three  real  coordinates.  Con- 
versely, any  three  real  numbers  x,  y,  z,  taken  as  x-,  y-,  and  z- 
coordinates,  respectively,  determine  a  point  P;  for  if  we  lay  otf  a 
distance  OA  =  x  on  the  X-axis,  OB=y  on  the   F-axis,  OC  =  z  on 

1 


COORDINATES 


[Chap.  I. 


Fig.  2. 


the  Z-axis,  and  draw  planes  through  A,  B,  C  parallel  to  the  co- 
ordinate planes,  these  planes  will  intersect  in  a  point  P  whose 
coordinates  are  x,  y,  and  z. 

It  will  frequently  be  more  convenient  to  determine  the  point 
P  whose  coordinates  are  x,  y,  and  z,  as  follows :  Lay  off  the 
distance  OA  =  x  on  the  X-axis  (Fig,  2).  From  A  lay  off  the 
distance  AD  =  ?/  on  a  parallel  to  the  F-axis.     From  D  lay  off  the 

distance  DP  =  2  on  a  parallel  to 
the  Z-axis. 

The  eight  portions  of  space 
separated  by  the  coordinate 
planes  are  called  octants.  If  the 
coordinates  of  a  point  P  are  a, 
'^  b,  c,  the  points  in  the  remaining 
octants  at  the  same  absolute 
distances  from  the  coordinate 
planes  are  (—  a,  b,  c),  (a,  —  b,  c), 
(a,  b,  -  c),  (-  a,-b,  c),  (-  a,  b,  -  c),  (a,  -  b,  -  c),  (-  a,  —  6,-  c). 
Two  points  are  symmetric  with  regard  to  a  plane  if  the  line 
joining  tliem  is  perpendicular  to  the  plane  and  the  segment 
between  them  is  bisected  by  the  plane.  They  are  symmetric  with 
regard  to  a  line  if  the  line  joining  them  is  perpendicular  to  the 
given  line  and  the  segment  between  them  is  bisected  by  the  line. 
They  are  symmetric  with  regard  to  a  point  if  the  segment  be- 
tween them  is  bisected  by  the  point. 

The  problem  of  representing  a  ligure  in  space  on  a  plane  is 
considered  in  descriptive  geometry,  where  it  is  solved  in  several 
ways  by  means  of  projections.  In  the  figures  appearing  in  this 
book  a  particular  kind  of  parallel  projection  is  used  in  which  the 
X-axis  and  the  Z-axis  are  represented  by  lines  perpendicular  to 
each  other  in  the  plane  of  the  paper ;  the  F-axis  is  represented  by 
a  line  making  equal  angles  with  the  other  two.  Distances 
parallel  to  the  X-axis  or  to  the  Z-axis  are  represented  correctly 
to  scale,  but  distances  parallel  to  the  F-axis  will  be  foreshortened, 
the  amount  of  which  may  be  chosen  to  suit  the  particular  drawing 
considered.  It  will  usually  be  convenient  for  the  student,  in 
drawing  figures  on  cross  section  paper,  to  take  a  unit  on  the 
y-axis  1/ V2  times  as  long  as  the  unit  on  the  other  axes. 


Art.  2]  ORTHOGONAL  PROJECTIONS  3 

EXERCISES 

1.  Plot  the  following  points  to  scale,  using  cross  section  paper  :  (1,  1,  1), 
(2,  0,  3),  (-  4,  -  1,  -4),  (-3,-4,  1),  (4,  4,  -  1),  (-7,  2,  3),  (-1,  6,  -6), 
(-4,2,8),  (3,  -4,  -1),  (2,1,  -3),  (-1,0,0),  (4,  -2,  2),  (0,  0,  2), 
(0,  -1,  0),  (-3,0,  0),  (0,  0,  0). 

2.  What  is  the  locus  of  a  point  for  which  x  =  0  ? 

3.  What  is  the  locus  of  a  point  for  which  x  =  0,  ?/  =  0  ? 

4.  What  is  the  locus  of  a  point  for  which  x  =  a,  y  =  b? 

5.  Given  a  point  {k,  I,  m),  write  the  coordinates  of  the  point  symmetric 
with  it  as  to  the  plane  XOY;  the  plane  ZOX;  the  X-axis;  the  F-axis;  the 
origin. 

2.  Orthogonal  projections.  The  orthogonal  projection  of  a 
point  oil  a  plane  is  the  foot  of  the  perpendicular  from  the  point 
to  the  plane.  The  orthogonal  projection  on  a  plane  of  a  segment 
PQ  of  a  line*  is  the  segment  P'Q\  joining  the  projections  P'  and 
Q'  oi  P  and  Q  on  the  plane. 

The  orthogonal  projection  of  a  point  on  a  line  is  the  point  in 
which  the  line  is  intersected  by  a  plane  which  passes  through  the 
given  point  and  is  perpendicular  to  the  given  line.  The  or- 
thogonal projection  of  a  segment  PQ  of  a  line  Z  on  a  second  line 
/'  is  the  segment  P'Q'  joining  the  projections  P'  and  Q'  of  P  and 
Q  on  I. 

For  the  purpose  of  measuring  distances  and  angles,  one  direc- 
tion along  a  line  will  be  regarded  as  positive  and  the  opposite 
direction  as  negative.  A  segment  PQ  on  a  directed  line  is 
positive  or  negative  according  as  Q  is  in  the  positive  or  nega- 
tive direction  from  P.  From  this  definition  it  follows  that 
PQ=-QP. 

The  angle  between  two  intersecting  directed  lines  I  and  V  will 
be  defined  as  the  smallest  angle  which  has  its  sides  extending 
in  the  positive  directions  along  I  and  V.  We  shall,  in  general, 
make  no  convention  as  to  whether  this  angle  is  to  be  considered 
positive  or  negative.  The  angle  between  two  non-intersecting 
directed  lines  I  and  V  will  be  defined  as  equal  to  the  angle  be- 
tween two  intersecting  lines  m  and  m'  having  the  same  directions 
as  I  and  V,  respectively. 

*  We  shall  use  the  word  line  throughout  to  mean  a  straight  line. 


COORDINATES 


[Chap.  I. 


Theorem  I.  The  length  of  the  projection  of  a  segment  of  a 
directed  line  on  a  second  directed  line  is  equal  to  the  length  of  the 
given  segment  midtiplied  by  the  cosine  of  the  angle  between  the  lines. 

Let  PQ  (Figs.  3  a,  3  b)  be  the  given  segment  on  I  and  let  P'Q' 
be  its  projection  on  V.  Denote  the  angle  between  I  and  I'  by  6. 
It  is  required  to  prove  that 

P' Q'  =  PQ  cos  0. 

Through  P'  draw  a  line  I"  having  the  same  direction  as  I.  The 
angle  between  V  and  I"  is  equal  to  6.     Let  Q"  be  the  point  in 


Fig.  3  a. 


Fig.  3  6. 


which  I"  meets  the  plane  through  Q  perpendicular  to  V.  Then 
the  angle  P'Q'Q"  is  a  right  angle.  Hence,  by  trigonometry, 
we  have 

P'Q'  =  P'Q"  cos  e. 

But  P'Q"  =  PQ. 

It  follows  that  P'Q'  =  PQ  cos  $. 

It  should  be  observed  that  it  makes  no  difference  in  this 
theorem  whether  the  segment  PQ  is  positive  or  negative.  The 
segment  PQ  =  r  will  always  be  regarded  as  positive  in  defining 


Theorem  II.  The  projection  on  a  directed  line  I  of  a  broken 
line  made  %ip  of  segments  P^P^,  P2P3,  ••',  Pn-\Pn  of  different  lines  is 
the  sum  of  the  projections  on  I  of  its  parts,  and  is  equal  to  the  pro- 
jection on  I  of  the  straight  line  P^Pn- 


Arts.  2,  3]       DIRECTION   COSINES   OF  A  LINE 


For,  let  P\,  P'o,  P'3,  •••,  P'„_i,  P'„  be  the  projections  of  P^,  P^, 
Ps,  ••-,  P„_i,  P„,  respectively.  The  sum  of  the  projections  is 
equal  to  P\P\, ;  that  is, 

P,P,  +  P,P,  +  ...  +  P'„_iP'„  =  P,P,, 

But  P'iP'„is  the  projection  of  PiP„.    The  theorem  therefore  follows. 

Corollary.  If  Pi,  P2,  •'•,  P„-i  «'"e  the  vertices  of  a  polygon,  the 
sum  of  the  projections  on  any  directed  line  I  of  the  segments  P^P^, 
PiP^i  •••>  Pn-\Pi  formed  by  the  sides  of  the  'polygon  is  zero. 

Since  in  this  case  P„  and  Pi  coincide,  it  follows  that  P\  and  P'„ 
also  coincide.     The  sum  of  the  projections  is  consequently  zero. 

EXERCISES 

1.  If  0  is  the  origin  and  P  any  point  in  space,  show  that  the  projections 
of  the  segment  OP  upon  the  coordinate  axes  are  equal  to  the  coordinates  of  P. 

2.  If  the  coordinates  of  Pi  are  Xi,  y^  ^i  and  of  Po  are  x-j,  2/2,  z-,,  show  that 
the  projections  of  the  segment  PiP^  upon  the  coordinate  axes  are  equal  to 
^•2  —  a^i,  2^2  —  2/i>  Zo  —  Z\,  respectively. 

3.  If  the  lengths  of  the  projections  of  PiP^  upon  the  axes  are  respectively 
3,  —  2,  7  and  the  coordinates  of  Pi  are  (-  4,  3,  2),  find  the  coordinates  of  P2. 

4.  Find  the  distance  from  the  origin  to  the  point  (4,  3,  12). 

5.  Find  the  distance  from  the  origin  to  the  point  (a,  h.  c). 

6.  Find  the  cosines  of  the  angles  made  with  the  axes  by  the  line  joining 
the  origin  to  each  of  the  following  points. 

(1,2,0)  (1,1,1)  (-7,6,2) 

(0,2,4)  (1,-4,2)  {:>-,iJ,z) 

3.    Direction  cosines  of  a  line.  kZ 

Let  I  be  any  directed  line  in 
space,  and  let  V  be  a  line  through 
the  origin  which  has  the  same 
direction.  If  «,  fi,  y  (Fig.  4) 
are  the  angles  which  V  makes 
with  the  coordinate  axes,  these 
are  also,  by  definition  (Art.  2), 
the  angles  which  I  makes  with 
the  axes.     They  are  called  the 

direction  angles  of  I  and  their  cosines  are  called  direction  cosines. 
The  latter  will  be  denoted  by  A,  /x,  v,  respectively. 


^ 

/,' 

^J^ 

h^. 

r/ 

-^/s 

a 

6  COORDINATES  [Chap.  I.  ' 

Let  P=(a,  b,  c)  be  any  point  on  I'  in  the  positive  direction  from 
the  origin  and  let  OP  =  r.     Then,  from  trigonometry,  we  have 

a  o      b  c 

\  =  cos  a  =    ,        /x  =  cos  p  =  -,         V  =  cos  y  =  -  • 
r  r  r 

Bnt  r  is  the  diagonal  of  a  rectangular  parallelepiped  whose  edges 

OA^a,         OB  =  b,         OG=c. 


Va^ 

-h62_,_c2 

h 

Va' 

'■  +  ¥  +  (? 

c 

Hence,  we  obtain  r  =  Va-  +  ^^  +  cl 

In  this  equation,  as  in  the  formulas  throughout  the  book,  except 
when  the  contrary  is  stated,  indicated  roots  are  to  be  taken  with 
the  positive  sign. 

By  substituting  this  value  of  r  in  the  above  equations,  we  obtain 

\  =  cos  a  = 


/x  =  cos  (3  = 


v=C0Sy  

Va"  +  b^+c' 

By  squaring  each  member  of  these  equations  and  adding  the 

results,  we  obtain  ,00. 

X--^  +  Ht--f-  v-'  =  l,  (1) 

hence  we  have  the  following  theorem. 

Theorem.  The  sum  of  the  squares  of  the  direction  cosines  of  a 
line  is  equal  to  unity. 

If  Ai,  fxi,  vi  and  Ao,  1^2,  v-^  are  the  direction  cosines  of  two  like 
directed  lines,  we  have 

Xl  =  A2,     fli  =  /Xo,     Vi  =  V2. 

If  the  lines  are  oppositely  directed,  we  have 

Ai  =  —  Ao,     /Ai  =  —  fJ.2,     Vi  =  —  Vo. 

4.  Distance  between  two  points.  Let  Pi  =(x\,  y^,  z^),  Pt  =  {x2,y2, 
Z2)  be  any  two  points  in  space.     Denote  the  direction  cosines  of  the 


Arts.  4,  5]    ANGLE  BETWEEN  TWO   DIRECTED   LINES       7 


line  P1P2  (Fig.  5)  by  X,  /a,  v  and  the  length  of  the  segment  P^Pz 
by  d.  The  projection  of  the  segment  P^P.,  on  each  of  the  axes  is 
equal  to  the  sum  of  the  projections 
of  P,0  and  OP.,  that  is 


Xd  =  X,  ~  x\,  fxd  =  2/2  —  Vn  vd  =  Zo  —  Zi. 

By  squaring  both  members  of  these 
equations,  adding,  and  extracting  the 
square  root,  we  obtain 


^1 


N^ 


N, 


Fig.  5. 


^M^ 


-rX 


a  =  V(a?a  -  ^1)'^  +  (:i/'2  -  Vi)'^ + («2  -  «i)2. 


(2) 


EXERCISES 

1.  Find  the  distance  between  (3,  4,  —  2)  and  (—  5,  1,  —  6). 

2.  Show  that  the  points  ( -  3,  2,  -  7),  (2,  2,  -  3),  and  (-  3,  6,  -  2)  are 
vertices  of  an  isosceles  triangle. 

3.  Show  that  the  points  (4,  3,  —  4),   (-  2,  9,   —  4),  and  (—  2,  3,  2)  are 
vertices  of  an  equilateral  triangle. 

■     4.    Express  by  an  equation  that  the  point  (x,  rj,  z)  is  equidistant  from 
(1,  1,  1)  and  (2,  3,  4). 

5.  Show  that  or^  +  y'^  +  z-  =  i  \s  the  equation  of  a  sphere  whose  center  is 
the  origin  and  whose  radius  is  2. 

6.  Find  the  direction  cosines  of  the  line  P1P21  given  : 


(a)  Pi  =  (0,  0,  0), 

(h)  P,=  (l,  1,  1), 

^(c)   Pi  =  (l,  -2,3), 


P2  =  (2,  3,  5). 
P2  =  (2,  2,  2). 
P2  =  (4,2,  -1). 


7.  What  is   known   about  the  direction   of    a  line    if   (a)  cos  a  =  0  ? 
(6)  cos  a  —  0  and  cos  /3  =  0  ?     (c)  cos  «  =  1  ? 

8.  Show  that  the  points  (3,  -  2,  7),  (6,  4,  —  2),  and  (5,  2,  1)  are  on  a 
line. 

9.  Find  the  direction  cosines  of  a  line  which  makes  equal  angles  with  the 
coordinate  axes. 

5.    Angle  between  two  directed   lines.      Let   li   and   l^  be   two 

directed  lines  having  the  direction  cosines  X,,  fXi,  vi  and  A2,  /X2,  V2, 
respectively.  It  is  required  to  find  an  expression  for  the  cosine 
of  the  angle  between  l^  and  I^.     Through  0  (Fig.  6)  draw  two 


COORDINATES 


[Chap.  I. 


^,' 


lines  OPi  and  OP^  having  the  same  di- 
rections as  li  and  l^,  respectively.  Let 
OP2  =  ^2  and  let  the  coordinates  of  P^  be 

x^  =  OM,        2/2  =  MN,        %<,  =  ArP2 

X  The  projection  of  OP^  on  OPi  is  equal 
to    the   sum  of  the   projections  of  the 
'  Fig.  6.  broken  line  OMNP.  on  OP^  (Art.  2). 

Hence  OP^  cos  0  =  03/ Ai  +  MN  y.^  +  ^^2  vi- 

<  But     OP2  =  »'2J      03/=  a;2  =  r2A.2J        ^^  =  2/2  =  ^*2/>t2)        -^-f*  =  ^2  =  ^2>'2- 


Hence,  we  obtain 


or 


r^  cos  ^  =  rjXiAa  +  >'2/'ii/^2  +  ^2»'i»'2) 

cos  e  =  \i\2  -}-  1*1(1.2  +  vivg. 


(3) 


The  condition  that  the  two  given    lines  are  perpendicular  is 
that  cos  ^  =  0.     Hence  we  have  the  following  theorem  : 

Theorem.      Two  lines  l^  and  Uwith  direction  cosines  Aj,  /xj,  vi  and 
X2,  /Ao)  V2,  respectively,  are  perpendicular  if 


'^l^a  +  t^lK-2  +''iv.2  =  0. 


(4) 


The  square  of  the  sine  of  6  may  be  found  from  (1)  and  (3). 
Since  sin^  ^  =  1  —  cos^  6,  it  follows  that 


sin2  6  =  (Ai^  +  ix^  +  v,2)  {X^  +ix,^  +  v.^)  -  (A1A2  +  M1M2  +  v,v,Y 

—  (Ai)a2  —  XofJ^xf  +  {lJ^iV2  —  M2Vl)^  +(viA2  -  V2Ai)2. 


<5.  Point  dividing  a  segment  in  a 
given  ratio.  Let  Pj  =  (.t„  .Vi,  z^)  and 
P2  =  (x2,  1/2,  Z2)  be  two  given  points 
(Fig.  7).  It  is  required  to  find  the 
point  P  =  (x,  y,  z)  on  the  line  P1P2 
such  that  P,P :  PP.  =  Wi :  m..  Let 
A,  iM,  V  be  the  direction  cosines  of 


(5) 


Fig  7. 


the  line  P1P2.    Then   (Art.  2,  Th.    I)  we  have 

P, P A  =  .T  —  Xi  and  PP2 \  =  X2—x. 
Hence  P^P  A  :  PP2  \  =  x  —  Xi  :  x^  —  x^  m^  :  wij. 


Art.  6]  POINT   DIVIDING  A  SEGMENT  9 

On  solving  for  x  we  obtain 

x  =  — ^-'— ! =,  (6) 

mi  +  m2 

c       1     1  „      m2!/i  +  mii/2 

Similarly,  y  = ; . 

~     mi  +  m^ 

It  should  be  noticed  that  if  vii  and  7?i2  have  the  same  sign,  P^P 
and  PPj  a.re  measured  in  the  same  direction  so  that  Plies  between 
Pj  and  Pj.  If  7Jii  and  ?/i2  have  opposite  signs,  P  lies  outside  the 
segment  PiPj.  By  giving  mj  and  7)12  suitable  values,  the  coor- 
dinates of  any  point  on  the  line  P1P2  can  be  represented  in 
this  way.  In  particular,  if  P  is  the  mid-point  of  the  segment 
P1P2,  vii  =  m^,  so  that  the  coordinates  of  the  mid-point  are 
_  a?!  +X2  _  Vx  +  ?/2  „  _  ^^  +Z2 


EXERCISES 

1.  Find  the  cosine  of  the  angle  between  the  two  lines  whose  direction 

cosines  are — ^,    — ^,    — ^ — -    and    — ;^,    — ^::,    — ^^- 
^14      \/l4      Vli  VSO      VSO      VSO 

2.  Find  the  direction  cosines  of  each  of  the  coordinate  axes. 

3.  The  direction  cosines  of  a  line  are  proportional  to  4,  —  3,  1.     Find 
their  values. 

4.  The  direction  cosines  of  two  lines  are  proportional  to  6,  2,  —  1  and 
—  3,  1,  —  5,  respectively.     Find  the  cosine  of  the  angle  between  the  lines. 

5.  Show  that  the  lines  whose  direction  cosines  are  proportional  to  3,  6, 
2  ;  —  2,  3,  —  6  ;  —  6,  2,  3  are  mutually  perpendicular. 

6.  Show  that  the  points  (7,  3,  4),  (1,  0,  6),  (4,  5,-2)  are  the  vertices 
of  a  right  triangle. 

7.  Show  that  the  points  (3,  7,  2),  (4,  3,  1),  (1,  6,  3),  (2,  2,  2)  are  the 
vertices  of  a  parallelogram. 

8.  Find  the  coordinates  of  the  intersection  of  the  diagonals  in  the  paral- 
lelogram of  Ex.  7. 

9.  Show  by  two  different    methods    that  the  three  points  (4,  13,  3), 
(3,  6,  4),  (2,  —  1,  5)  are  coUinear. 


10 


COORDINATES 


[Chap.  I. 


10.  A  line  makes  an  angle  of  75°  with  the  A'-axis  and  30°  with  the  F-axis. 
How  many  positions  may  it  have  ?  Find,  for  each  position,  the  cosine  of  the 
angle  it  makes  with  the  Z-axis. 

11.  Determine  the  coordinates  of  the  intersection  of  the  medians  of  the 
triangle  witli  vertices  at  (1,  2,  3),  (2,  3,  1),  (3,  1,  2). 

12.  Prove  that  the  medians  of  any  triangle  meet  in  a  point  twice  as  far 
from  each  vertex  as  from  the  mid-point  of  the  opposite  side.  This  point  is 
called  the  center  of  gravity  of  the  triangle. 

"  13.  Prove  that  the  three  straight  lines  joining  the  mid-points  of  oppo- 
site edges  of  any  tetrahedron  meet  in  a  point,  and  are  bisected  by  it.  This 
point  is  called  the  renter  of  gravity  of  the  tetrahedron. 

14.  Show  that  the  lines  joining  each  vertex  of  a  tetrahedron  to  the  point 
of  intersection  of  the  medians  of  the  opposite  face  pass  through  the  center  of 
gravity. 

15.  Show  that  the  lines  joining  the  middle  points  of  the  sides  of  any 
quadrilateral  form  a  parallelogram. 

16.  Show  how  the  ratio  mi  :  ??i2  (Art,  6)  varies  as  P  describes  the  line 
P1P2. 


7.    Polar  Coordinates.     Let  OX,  0  Y,  OZ  be  a  set  of  rectangular 
axes  and  P  be  any  point  in  space.     Let  OP  =  p  have  the  direc- 


AZ 


tion  angles  a,  ^,  y.  The  position 
of  the  line  OP  is  determined  by 
rt,  (3,  y  and  the  position  of  P  on 
the  line  is  given  by  p,  so  that  the 
position  of  the  point  P  in  space 
is  fixed  when  p,  a,  ^,  y  are 
known.  These  quantities  p,  a,  ft, 
y  are  called  the  polar  coordinates 
Y  ^^i"-  «■  of  P.     As   a,  ft,  y  are  direction 

angles,  they  are  not  independent,  since  by  equation  (1) 

cos^  a  +  cos^  ft  -f  cos^  y  =  1. 

If  the  rectangular  coordinates  of  P  are  x,  y,  z,  then  (Art.  3) 

X  =  p  cos  a,         y  —  p  cos  ft,         z  =  p  cos  y. 

8.  Cylindrical  coordinates.  A  point  is  determined  when  its 
directed  distance  from  a  fixed  plane  and  the  polar  coordinates  of 
its  orthogonal  projection  on  that  plane  are  known.  These  co- 
ordinates are  called  the  cylindrical  coordinates  of  a  point.     If  the 


Arts.  8,  9] 


SPHERICAL  COORDINATES 


11 


point  P  is  referred  to  the  rectangular 
axes  X,  y,  z,  and  the  fixed  plane  is  taken 
as  2  =  0  and  the  a;-axis  for  polar  axis, 
we  may  write  (Fig.  9) 

x=  p  cos  6,         y  =  p  sin  6,         z  =  z, 

in  which  p,  0,  z  are  the  cylindrical  coordi- 
nates of  P. 

9.  Spherical  coordinates.  Let  OX,  0  Y,  OZ,  and  P  be  chosen  as 
in  Art.  7,  and  let  P  be  the  orthogonal  projection  of  P  on  the  plane 
XOY.  Draw  OP.  The  position  of  P  is  defined  by  the  distance 
p,  the  angle  </>=  ZOP  wliich  the  line  OP  makes  with  the  2;-axis, 
and  the  angle  6  (measured  by  the  angle  XOP)  which  the  plane 
through  P  and  the  2;-axis  makes  witli  the  plane  XOZ.  The  num- 
bers p,  ^,  6  are  called  the  spherical  coordinates  of  P.  The  length 
p  is  called  the  radius  vector,  the  angle  <{>  is  called  the  co-latitude, 

and  6  is  called  the  longitude. 

If    P  =  (x,   y,   z),    then,   from    the   figure 

(Fig.  10), 

OP  =  p  cos  (90  -  <^)  =  p  sin  <^. 
Hence      x  =  p  sin  cf>  cos  8, 
y  =  P  sin  <^  sin  6, 
FiG.  10.  '  z  =  p  cos  <^. 

On  solving  these  equations  for  p,  <fi,  6,  we  find 

0  =  arc  tan  "  • 


p  =  Vic^  +  2/^  +  2;S       <^  =  arc  cos 


Va-2  +  ?/2  -f-  ;32 


EXERCISES 

1.  What  locus  is  defined  by  p  =:  1  ? 

2.  What  locus  is  defined  by  a  =  60°  ? 

3.  What  locus  is  defined  by  ^  =  30°  ? 

4.  What  locus  is  defined  by  ^  =  45°  ? 

5.  TransforLu  x'^ -\- y'^  +  z- — '^  to:  (a)  polar  coordinates,  {h)  spherical 
coordinates,  (c)  cylindrical  coordinates. 

6.  Transform  x^  -f  2/^  =  z'^  into   spherical   coordinates  ;   into   cylindrical 
coordinates. 

7.  Express  the  distance  between  two   points   in   terras   of    tlieir   polar 
coordinates. 


CHAPTER  II 

PLANES   AND   LINES 

10.   Equation  of  a  plane.      A   plane   is    characterized   by   the 

properties : 

(a)  It  contains  three  points  not  on  a  line. 

(b)  It  contains  every  point  on  any  line  joining  two  points  on  it. 

(c)  It  does  not  contain  all  the  points  of  space. 

Theokem.  The  locus  of  the  points  ivhose  coordinates  satisfy  a 
linear  equation 

Ax  +  By+  Cz  +  D  =  0  (1) 

with  real  coefficients  is  a  plane. 

We  shall  prove  this  theorem  on  the  supposition  that  C  ^  0. 
Since  A,  B,  C  are  not  all  zero,  a  proof  for  the  case  in  which 
C—0  can  be  obtained  in  a  similar  way. 

It  is  seen  by  inspection  that  the  coordinates  (  0,  0,  —  77  ), 
fo,  1,  _(-S  +  ^)\    f-^^^  Q^  _  (A  +  D)\  gatisfy  the  equation.     These 

three  points  are  not  collinear,  since  no  values  of  m,,  m2  other  than 
zero  satisfy  the  simultaneous  equations  (Art.  6) 

m,  =  0,         nu  =  0,         m^A  +  ni2B  =  0. 

Let  Pi  =  (.i\,  iji,  Zi)  and  P,  =  {x^,  y^,  z.,)  be  any  two  points  whose 
coordinates  satisfy  (1).  The  coordinates  of  any  point  P  on  the 
line  P1P2  are  of  the  form 

miX2  +  m^Xi  _  /«.,//.j  +  nuV\  ^  _  tn^z^  +  n^z^ 

X  — f         y  — —J         z  —  ■  • 

wii  +  m.>  '  m^  +  m.2  m^  +  mj 

The  equation  (1)  is  satisfied  by  the  coordinates  of  P  if 

nu{Ax,  +  By,  -|-  Cz,  +  D)-\-  m,{Ax.,  +  By.,  +  Cz^  +  Z))  =  0, 

but  since  the  coordinates  of  I\  and  P^  satisfy  (1),  we  have 
Ax,  +  By,  +  Cz,-\-D  =  0,         Ax,  4-  By,  +  Cz,  +  D  =  0, 

hence  the  coordinates  of  P  satisfy  (1)  for  all  values  of  m,  and  wij. 

12 


Arts.  11,  12]      INTERCEPT   FORM   OF   THE   EQUATION      13 

Finally,  not  all  the  points  of  space  lie  on  the  locus  defined  by 
(1),  since  the  coordinates  [  0,  0,  —  ^ — ^ — ^  j  do  not  satisfy  (1). 
This  completes  the  proof  of  the  theorem. 

11.  Plane  through  three  points.  Let  (ic,,  y^,  z^),  (x^,  n^,  z^, 
(^3,  ?/3,  Zj)  be  the  coordinates  of  three  non-collinear  points.  The 
condition  that  these  points  all  lie  in  the  plane 

^x  +  -B^  +  Cz  +  Z>  =  0 

is  that  their  coordinates  satisfy  this  equation,  thus 

Ax^  +  %i  +  C^i  +  i>  =  0, 
Ax^  +  By^_  +  Cz2+  D=:  0, 
Ax,  +  By,  +  Cz,  +  D  =  0. 

The  condition  that  four  numbers  A,  B,  C,  D  (not  all  zero) 
exist  which  satisfy  the  above  four  simultaneous  equations  is 


X      y      z      1 

.-c,     ?/i     Zi     1 


=  ^-^yL^.^.2..c.^Ml 


^1        2/2        ^2 

•^z     Vz     ^3 

This  is  the  required  equation,  for  it  is  the  equation  of  a  plane, 
since  it  is  of  first  degree  in  x,  y,  z  (Art.  10).  The  plane  passes 
through  the  given  points,  since  the  coordinates  of  each  of  the  given 
points  satisfy  the  equation. 

12.  Intercept  form  of  the  equation  of  a  plane.  If  a  plane  inter- 
sects the  X-,  Y-,  Z-axes  in  three  points  '^1,  B,  C,  respectively,  the 
segments  OA,  OB,  and  OC  are  called  the  intercepts  of  the  plane. 
Let  A,  B,  C  all  be  distinct  from  the  origin  and  let  the  lengths  of 
the  intercepts  be  a,  b,  c,  so  that  A  =(a,  0,  0),  B  =(0,  b,  0),  C  = 
(0,  0,  c).  The  equation  (2)  of  the  plane  determined  by  these  three 
points  (Art.  11)  may  be  reduced  to 

^  +  f-f-  =  l.  (3) 

a     b      c 

This  equation  is  called  the  intercept  form  of  the  equation  of  a 
plane. 


14 


PLANES  AND   LINES 


[Chap.  IL 


EXERCISES 

1.    Find  the  equation  of  the  plane  through  the  points  (1,  2,  3),  (3,  1,  2), 
(5,  -  1,  3). 

•*    2.    Find  the  e(iuation  of  the  plane  through  the  points  (0,  0,  0),  (1,  1,  1), 
(2,  2,  -  2) .     What  are  its  intercepts  ? 

3.  Prove  that  the  four  points  (1,  2,  3),   (2,  4,  1),   (-  1,  0,  1),   (0,  0,  5) 
lie  in  a  plane.     Find  the  equation  of  the  plane. 

4.  Determine  k  so  that  the  points  (1,  2,  -  1),  (3,  -  1,  2),  (2,  -  2,  3), 
(1,  —  1,  k)  shall  lie  in  a  plane. 

'     5.    P'ind   the   point   of   intersection  of   the  three   planes,    a:  +  ?/  +  2  =  6, 
22-y+2a!  =  0,  x-2y  +  33;  =  4. 

13.    The  normal  form  of   the   equation    of   a   plane.     Let  ABC 

(Fig.  11)  be  any  plane.  Let  OQ  be  drawn  through  the  origin  per- 
pendicular to  the  given  plane 
and  intersecting  it  at  P'.  Let 
the  direction  cosines  of  OQ,  be 
A,  fi,  V  and  denote  the  length  of 
the  segment  OP  by^:*. 

Let  P  =  {x,  y,  z)  be  any  point 
in  the  given  plane.  The  projec- 
tion of  P  on  OQ  is  P'  (Art.'  2). 
Draw  OP  and  the  broken  line 
OMNP,  made  up  of  segments 
CM  =  X,  MN  =  y,  and  NP  =  z, 


Fio.  11. 


parallel  to  the  X-,  Y-,  and  Z-axes,  respectively.  The  projections  of 
OP  and  OMNP  on  OQ  are  equal  (Art.  2,  Th.  II).  The  projection 
of  the  broken  line  is  \x  +  fxy  +  vz,  the  projection  of  OP  is  OP'  ov  jy, 
so  that 

Xx  +  ixy  +  vz=2^.  (4) 

This  equation  is  satisfied  by  the  coordinates  of  every  point  P  in 
the  given  plane.  It  is  not  satisfied  by  the  coordinates  of  any 
other  point.  For,  if  Pj  is  a  point  not  lying  in  the  given  plane,  it 
is  similarly  seen,  since  the  projection  of  OPi  on  OQ  is  not  equal  to 
p,  that  the  coordinates  of  Pi  do  not  satisfy  (4). 

Hence,  (4)  is  the  equation  of  the  plane.  It  is  called  the  normal 
form  of  the  equation  of  the  plane.  The  number  p  in  this  equa- 
tion is  positive  or  negative,  according  as  P'  is  in  the  positive  or 
negative  direction  from  0  on  OQ. 


Art.  14]  REDUCTION   OF  THE   EQUATION  15 

14.    Reduction  of  the  equation  of  a  plane  to  the  normal  form.     Let 
Ax  +  By+Cz  +  D  =  0  (5) 

be  any  equation  of  first  degree  with  real  coefficients.  It  is  required 
to  reduce  this  equation  to  the  normal  form.  Let  Q  =  (^4,  B,  G) 
be  the  point  whose  coordinates  are  the  coefficients  of  x,  y,  z  in  this 
equation.  The  direction  cosines  of  the  directed  line  OQ  are 
(Art.  3) 

X  ^  B  C  ... 

Va^^+W+c^  V^2  +  B'  +  C^         Va-"  +  ^  +  (72 

If  we  transpose  the  constant  term  of  (5)  to  the  other  member  of 
the  equation,  and  divide  both  members  by  y/ A^  +  B'^  +  C^,  we 
obtain 

A  ,  B 

■X  +  —Z  -y 


^     ,         ^  .=  -^  •  (7) 

The  plane  determined  by  (7)  is  identical  with  that  determined 
by  (5)  since  the  coordinates  of  a  point  will  satisfy  (7)  if,  and  only 
if,  they  satisfy  (5).  By  subtituting  from  (6)  in  (7)  and  comparing 
with  (4),  we  see  that  the  locus  of  the  equation  is  a  plane  perpen- 
dicular to  OQ,  and  intersecting  OQ  at  a  point  P'  whose  distance 
from  0  is 

P=      ^  ~^    ='  (8) 

V^2  +  B^  +  C^ 

In  these  equations,  the  radical  is  to  be  taken  with  the  positive 
sign.  The  coefficients  of  x,  y,  z  are  proportional  to  A,  /a,  v  in  such 
a  way  that  the  direction  cosines  of  the  normal  to  the  plane  are 
fixed  when  the  signs  of  A,  B,  C  are  known.  But  the  plane  is  not 
changed  if  its  equation  is  multiplied  by  —  1,  hence  the  position 
of  the  plane  alone  is  not  sufficient  to  determine  the  direction  of 
the  normal.  In  order  to  define  a  positive  and  a  negative  side  of 
a  plane  we  shall  first  prove  the  following  theorem: 

Theorem.  Tico  points  Pi,  P^  are  on  the  same  side  or  on  opposite 
sides  of  the  plane  Ax  -\-  By +  Cz-\-  D  =  0,  according  as  their  coordi- 
nates make  the  first  member  of  the  equation  of  the  plane  have  like  or 
unlike  signs. 


16 


PLANES  AND  LINES 


[Chap.  IL 


For,  let  Pi=(xi,  y^,  z-^),  Po  =  (x2,  y-i,  x^)  be  two  points  not  lying 
on  the  plane.  The  point  P  =  {x,  y,  z)  in  which  the  line  PiPo  inter- 
sects the  plane  is  determined  (Art.  6)  by  the  values  of  mi,  m^ 
which  satisfy  the  equation 

my{Ax^  +  By.  -f-  Cz^  +  Z>)  +  m-lAxi  +  By^  +  Cz^  +  D)  =  0. 

If  Axi  +  7?//i  +  Czi  +  D  and  Ax.  +  By.  +  Cz.  +  D  have  unlike 
signs,  then  m^  and  iiu  have  the  same  sign,  and  the  point  P  lies  be- 
tween Pi  and  P.2.  If  Axi  -f-  £//i  +  Cz^  -f-  Z>  and  yl.r.  +  By.  -\-  Cz^ 
+  D  have  the  same  sign,  then  the  numbers  m^,  m^  have  opposite 
signs,  hence  the  point  P  is  not  between  Pj  and  Pg. 

When  all  the  terms  in  the  equation 

Ax  +  By  +  Cz-\-D  =  0 

are  transposed  to  the  first  member,  a  point  (x^,  _?/„  Zi)  will  be  said 
to  be  on  the  positive  side  of  the  plane  if  Axi  +  By^  +  Cz^  +  Z)  is  a 
positive  number;  the  point  will  be  said  to  be  on  the  negative  side 
if  this  expression  is  a  negative  number.  Finally,  the  point  is  on 
the  plane  if  the  expression  vanishes.  It  should  be  observed  that 
the  equation  must  not  be  multiplied  by  —  1  after  the  positive  and 
negative  sides  have  been  chosen. 

15.  Angle  between  two  planes.  The  angle 
between  two  planes  is  equal  to  the  angle 
between  two  dii-ected  normals  to  the  planes  ; 
hence,  by  Arts.  5  and  14,  we  have  at  once 
the  following  theorem : 


Theorem.     Tlie  cosine  of  the  angle  6  be- 
tween two  planes 

Ax  -{-By-\-Cz  +  D  =  0, 
A'x  +  B'y  +  C'z  +  D'  =  0 

is  defined  by  the  equation 

A  A'  +BB'-irCC' 


Fm.  12. 


COS  6  = 


z=:'  (9) 

In  particular,  the  condition  that  the  planes  are  perpendicular  is 
AA'  4-  BB'  +  CC  =  0.  (10) 


Arts.  15,  16] 

DISTANCE   TO  A  POINT 

FROM 

A  PLAl 

The  conditions  that  the  planes  ; 

ire  parallel  are 

(Art.  3) 

A 

B 

C 

A' 

B' 

C 

17 


(11) 

The  equations  (11)  are  satisfied  whether  the  normals  have  the 
same  direction  or  opposite  directions.  From  the  definition  of  the 
angle  between  two  planes  it  follows  that  in  the  first  case  the  two 
planes  are  parallel  and  in  the  second  case  they  make  an  angle  of 
180  degrees  with  each  other.  We  shall  say,  however,  that  the 
planes  are  parallel  in  each  case. 

16.    Distance  to  a  point  from  a  plane.     Let  P  =  (x^,  y^,  z^)  be  a 

given  point  and  Ax  +  B>/  +  Cz  +  D  =  0  be  the  equation  of  a  given 
plane.     The  distance  to  P  from  the  plane  is  equal  to  the  distance 
from  the  given  plane  to  a  plane  through  P  parallel  to  it. 
The  equation 

Ax  +  By+Cz-  (Ax,  +  By,  +  Cz,)  =  0 

represents  a  plane,  since  it  is  of  first  degree  with  real  coefficients 
(Art.  10).  It  is  parallel  to  the  given  plane  by  Eqs.  (11).  It  passes 
through  P  since  the  coordinates  of  P  satisfy  the  equation.  When 
the  equations  of  the  planes  are  reduced  to  the  normal  form,  they 
become,  respectively, 

A  ,  B 


C  -D 


A  ,  B 

—  — x-\ y 

V^2  _^  ^  4.  (72         y-^42  +  £2  ^  (72 

I  C  .^  ^1-^1  +  %i  +  Cz,  _ 

^W+W+~C^        VA^  +  -B2  +  02 

The  second  members  of  these  two  equations  represent  the  dis- 
tances of  the  two  planes  from  the  origin,  hence  the  distance  from 
the  first  plane  to  the  second,  which  is  equal  to  the  distance  d  to  P 
from  the  given  plane,  is  found  by  subtracting  the  former  from  the 
latter. 


18  PLANES  AND   LINES  [Chap.  IL 

The  result  is  ^       ,    n       ,  ^      ,    r. 

^  ^  Axt  +  By  I  +  Czi+D  _  .^2) 

Va'^  +  B-  +  C^ 

The  direction  to  P  from  the  plane,  along  the  normal,  is  positive 
or  negative  according  as  the  expression  in  the  numerator  of  the 
second  member  is  positive  or  negative  (Art.  14),  that  is,  according 
as  P  is  on  the  positive  or  negative  side  of  the  plane. 

EXERCISES  / 

1.  Reduce  the  equation  3  x  —  12  ?/  —  4  z  —  26  =  0  to  the  normal  form. 

2.  Write  the  equation  of  a  plane  through  the  origin  parallel  to  the  plane 
X  +  2  y  =  6. 

3.  What  is  the  distance  from  the  plane  3x  +  4y  —  z  =  5  to  the  point 
(2,2,2)? 

4.  Find  the  distance  between  the  parallel  planes 

2x  —  i/  +  32  =  4,     -Ix-y  +  Zz  +  b  =0. 

5.  Which  of  the  points  (4,  3,  1),  (1,  -4,  3),  (3,  5,  2),  (-  1,  2,  -2), 
(5,  4,  6)  are  on  the  same  side  of  the  plane  5x  —  2y  —  32  =  0  as  the  point 
(1,  6,  -  3)  ? 

6.  Find  the  coordinates  of  a  point  in  each  of  the  dihedral  angles  formed 
by  the  planes 

3x  +  2y  +  5s-4  =  0,    x-2?/-2;  +  6  =  0. 

7.  Show  that  each  of  the  planes  25  x  +  39  ?/  +  8  2  —  43  =  0  and  25  x 
—  .39?/  -|-  112  2  +  113  =  0  bisect  a  pair  of  vertical  dihedral  angles  formed  by 
the  planes  o  x  +  12  2  +  7  =  0  and  3«/  —  42  —  6  =  0,  Which  plane  bisects 
the  angle  in   which  the  origin  lies  ? 

8.  Find  the  equation  of  the  plane  which  bisects  that  angle  formed  by 
the  planes  3x  —  22/  +  2  —  4  =  0,  2x+2/  —  82  —  2  =  0,  in  which  the  point 
(1,  3,  -2)  lies.  -;  _  ' 

9.  Find  the  equations  of  the  planes  which  bisect  the  dihedral  angles 
formed  by  the  planes  AxX  +  Biy  +  dz  +  Di  =  0,  A-^x  +  B-iy  +  C-iZ  +  D-2  =  0, 

10.  Find  the  equation  of  the  locus  of  a  point  whose  distance  from  the 
origin-is  equal  to  its  distance  from  the  plane  3x  +  2/  —  22  =  11. 

11.  Write  the  equation  of  a  plane  whose  distance  from  the  point  (0,  2,  1) 
is  3,  and  which  is  perpendicular  to  the  radius  vector  of  the  point  (2,  —  1,-1). 

12.  Show  that  the  planes  2x-j/  +  2  +  3  =  0,  x-?/  +  42  =  0,  Zx  +  y 
-22  +  8  =  0,  4x-2«/  +  22-5  =  0,  9x  +  3«/-62  —  7=0,  and  lx-1  y 
+  28  2  —  6  =  0  bound  a  parallelopiped. 

13.  Write  the  equation  of  a  plane  through  (1,  2,  —  1),  parallel  to  the 
plane  x  —  2j/  —  2  =  0,  and  find  its  intercepts. 


Arts.  17,  18]        DIRECTION  COSINES  OF  THE  LINE  19 

14.  Find  the  equation  of  the  plane  passing  through  the  points  (1,  2,  3), 
(2,  —  3,  6)  and  perpendicular  to  the  plane  4x  +  2y  +  3z  =  l. 

15.  Find  the  equation  of  the  plane  through  the  point  (1,  3,  2)  pei-pen- 
dicular  to  the  planes 

2x  +  Sy-iz  =  2,    ix-Sy-2z  =:5. 

16.  Show  that  the  planes  x  +  2y  —  z  =  0,  y  +  1  z  —  2  =  0,  x  —  2y  —  z 
—  4  =  0,  X  +  Sy  +  z  =  -i,  and  Sx  +  Sy  —  z  —  8  bound  a  quadrilateral 
pyramid. 

17.  Find  the  equation  of  the  locus  of  a  point  which  is  3  times  as  far 
from  the  plane  3x  —  6y  —  2z  =  0  as  from  the  plane  2x  —  y  +  2z  =  9. 

18.  Determine  the  value  of  m  such  that  the  plane  mx  +  2y  —  Sz  —  14i 
shall  be  2  units  from  the  origin. 

19.  Determine  k  from  tlie  condition  that x  —  ky  +  Sz  —  2  shall  be  perpen- 
dicular to  3  X  +  4  y  —  2  z  =  5. 

1 7.  Equations  of  a  line.  Let  A^x  +  B^y  +  C^z  +  D^  =  0  and  A2X 
-f-  B^y  +  C2Z  -\-  0-2  =  0  he  the  equations  of  two  non-parallel  planes. 
The  locus  of  the  two  equations  considered  as  simultaneous  is  a 
line,  namely,  the  line  of  intersection  of  the  two  planes  (Art.  10). 
The  simultaneous  equations 

A,x  +  B,y  +  C,z  +  A  =  0, 

A2X  +  Biy  +  C2Z  +  D2  =  0 

are  called  the  equations  of  the  line. 

The  locus  represented  by  the  equations  of  two  parallel  planes, 
considered  as  simultaneous,  will  be  considered  later  (Art.  33). 

18.  Direction  cosines  of  the  line  of  intersection  of  two  planes. 
Let  A,  fx,  V  be  the  direction  cosines  of  the  line  of  intersection  of 
the  two  planes 

ii  =  A,x  +  B,y  -f-  C,z  -f  Z>i  =  0, 
L2  =  -420;  +  B2y  +  C2Z  -f  A  =  0- 

Since  the  line  lies  in  the  plane  A  =  0,  it  is  perpendicular  to  the 
normal  to  the  plane.     Hence,  (Arts.  5,  1^ 

\A,  +  fjiB,  +  vCi  =  0. 
Similarly,  XA.  +  fxBz  +  vC'j  =  0. 

By  solving  these  two  equations  for  the  ratios  of  A,  /u,,  v,  we  obtain 


^ fjt. 


B,C2  -  B2C,      C,A2  -  C2A,     A.B.  -  A2B, 


(13) 


20  PLANES  AND  LINES  [Chap.  IL 

The  denominators  in  these  expressions  are,  therefore,  proportional 
to  the  direction  cosines.  In  many  problems,  they  may  be  used 
instead  of  the  direction  cosines  themselves,  but,  in  any  case,  the 
actual  cosines  may  be  determined  by  dividing  these  denominators 
by  the  square  root  of  the  sum  of  their  squares.  It  should  be 
observed  that  the  equations  of  a  line  are  not  sufficient  to  deter- 
mine a  positive  direction  on  it. 

19.  Forms  of  the  equations  of  a  line.  If  A,  /u,,  v  are  the  direction 
cosines  of  a  line,  and  if  P,  ={xi,  y^,  Zj)  is  any  point  on  it,  the 
distance  d  from  Pj  to  another  point  P  =  (x,  y,  z)  on  the  line  satis- 
fies the  relations  (Art.  4) 

Xd  =  X  —  Xi,  fxd  =  y  —  yi,  vd  =  z  —  z^. 
By  eliminating  d,  we  obtain  the  equations 

\  fJi  V 

which  are  called  the  symmetric  form  of  the  equations  of  the  line. 
Instead  of  the  direction  cosines  themselves,  it  is  frequently 
convenient  to  use,  in  these  equations,  three  numbers  a,  b,  c,  pro- 
portional, respectively,  to  A,  /a,  v.     The  equations  then  become 


«i  _  ?/  -  .Vi  _  ^ 


(15) 


a  b  c 

They  may  be  reduced  to  the  preceding  form  by  dividing  the  de- 
nominator of  each  member  by  V«^  +  ^^  +  c^  (Art.  3). 

If  the  line  (15)  passes  through  the  point  P.;,={x2,  y^,  z^,  the 
coordinates  of  P^  satisfy  the  equations,  so  that 

a  b  c 

On   eliminating  a,  b,  c   between   these   equations   and   (15),  we 
obtain 

x-xi  ^  y-yi  ^  z-Zi  ^  ^^q-^ 

x^-xi     yt-y,     zi-zx 

These  equations  are  called  the  two-point  form  of  the  equations 
of  a  line. 


Art.  20]  PARAMETRIC   EQUATIONS  OF  A  LINE  21 

20.  Parametric  equations  of  a  line.  Any  point  on  a  line  may 
be  defined  in  terms  of  a  fixed  point  on  it,  the  direction  cosines  of 
the  line,  and  the  distance  d  of  the  variable  point  from  the  fixed 
one.     Thus,  by  Art.  4 

x=:Xi  +  Xd,   yz=y^^  fid,   z  =  z^  +  vd.  (17) 

If  A,  /A,  V  are  given  and  (x^,  y^,  z^)  represents  a  fixed  point,  any 
point  {x,  y,  z)  on  the  line  may  be  defined  in  terms  of  d.  To  every 
real  value  of  d  corresponds  a  point  on  the  line,  and  conversely. 
These  equations  are  called  parametric  equations  of  the  line,  the 
parameter  being  the  distance. 

It  is  sometimes  convenient  to  express  the  coordinates  of  a 
point  in  terms  of  a  parameter  k  which  is  defined  in  terms  of  d  by 
a  linear  fractional  equation  of  the  form 

y  +  8k 
in  which  a,  /3,  y,  8  are  constants  satisfying  the  inequality 

«8  -  ^y  ^  0. 
By  substituting  these  values  of  d  in  (17)  and  simplifying,  we 
obtain  equations  of  the  form 

in  which  a^,  b^,  etc.,  are  constants.  Equations  (18)  are  called  the 
parametric  equations  of  the  line  in  terms  of  the  parameter  k. 

It  should  be  observed  that  the  denominators  in  the  second 
members  of  equations  (18)  are  all  alike.  Each  value  of  k  for 
which  a^  -\-b^K=^0  determines  a  definite  point  on  the  line.  As 
«4  +  b^K  approaches  zero,  the  distance  of  the  corresponding  point 
from  the  origin  increases  without  limit.  To  the  value  deter- 
mined by  tti  4-  64K  =  0  we  shall  say  that  there  corresponds  a 
unique  point  which  we  shall  call  the  point  at  infinity  on  the  line. 

EXERCISES 

1.  Fiud  the  points  in  which  the  following  lines  pierce  the  coordinate 
planes  : 

(a)  X +2y -Sz  =  1,     3x-2y  +  5z  =  2. 

(b)  x  +  Sy  +  bz  =  0,     5x-Sy  +  z  =  2. 

(c)  x  +  2y-5  =  0,        2x-Sy  +  2z  =  T. 


22  PLANES  AND   LINES  [Chap.  IL 

2.  Write  the  equations  of  the  line  x  +  y  —  3  s  =  Cr,  2  x  —  y  -\-  2z  =  1  in 
the  symmetric  form,  the  two-point  form,  the  parametric  form. 

3.  Show  that  the  lines  4a;  +  2/  —  3^  =  0,  2x  —  y  +  2z  +  Q  —  0,  and  8 x 
—  y  -\-  z  =  1,  lOx  +  2/  —  •l^  +  lrrO  are  parallel. 

4.  Write  the  equations  of  the  line  through  (3,  7,  3)  and  (—  1,  5,  6). 
Determine  its  direction  cosines. 

5.  Find  the  equation  of  the  plane  passing  through  the  point  (2,  —  2,  0) 
and  perpendicular  to  the  line  2  =  3,  ?/  =  2  .r  —  4. 

6.  Find  the  value  of  k  for  which  the  lines  ^^^  =  ^-±-i  =  ^^^  and  ^—^ 

2  k        k+l        3  3 

y  +  5      r  +  2  T      1 

=  ^— ! — =  — ! —  are  perpendicular. 
1  ^•-2 

7.  Do  the  points  (2,  4,  6),  (4,  6,  2),  (1,  3,  8)  lie  on  a  line  ? 

8.  For  what  value  of  k  are  the  points  {k,  —  3,  2),  (2,  —  2,  3),  (fi,  —1,  4) 
coUinear  ? 

9.  Is  there  a  value  of  k  for  which  the  points  (k,  2,  —  2),  (2,  —  2,  A-),  and 
(—2,  1,  3)  are  coUinear? 

10.  Show  that  the  line'  ^^—^  =  ^-^  =  ?-^  lies  in  the  plane  2x+2y 

3-14 

-0  +  3  =  0. 

11.  In  equations  (18)  show  that,  as  k  approaches  infinity,  the  correspond- 
ing point  approaches  a  definite  point  as  a  limit.  Does  this  limiting  point  lie 
on  the  given  line  ? 

21.    Angle  which  a  line  makes  with  a  plane.     Given  the  plane 
Ax  +  By  ^  Cz  +  D  =  0 

and  the  line  ^^^=^  =  -"^^^^  =  ^-^^^. 

a  b  c 

The  angle  which  the  line  makes  with  the  plane  is  the  complement 
of  the  angle  which  it  makes  with  the  normal  to  the  plane.  The 
direction  cosines  of  the  normal  to  the  plane  are  proportional  to 
A,  B,  C  and  the  direction  cosines  of  the  line  are  proportional'  to 
a,  b,  c,  hence  the  angle  6  between  the  plane  and  the  line  is  de- 
termined (Art.  5)  by  the  formula 

sin  6  =  — —  — '  (19) 

V^2  +  B^+  C'2  Va2  +  62  +  c2 


Arts.  21,  22]     DISTANCE  FROM  A  POINT  TO  A  LINE       23 

EXERCISES 

1.  Show  that  the   planes   2x  -Sy  +  z  +  \  =0,  5x  +  z  —  l  =0,  ix  + 
9y  —  z  —  5  =  0  have  a  line  in  common,  and  find  its  direction  cosines. 

2.  Write  the  equations  of  a  line  which  passes  through  (5,  2,  6)  and  is 
parallel  to  the  line  2  x  —  3  z  +  y  —  2  =  0,  x  +  y  +  z  +  l=0. 

3.  Find  the  angle  which  the  line  x  +  y  +  2z  —  0,  2x—  y  +  2z  —  1=0 
makes  with  the  plane  Sx  +  6z  —  5y  +  l  =0. 

4.  Find  the  equation  of  the  plane  through  the  point    (2,  —  2,  0)  and 
peqiendicular  to  the  line  x  +  2y  —  Sz  =  i,  2x  —  Sy  +  4iZ  =  0. 

5.  Find   the   equation  of   the   plane   determined  by  the   parallel   lines 
X  +  I  _  y  —  2  _  z    X  —  .3  _  y  +  4  _  z  —  I 

3~2~r       3~2~1' 

6.  For  what  value  of  k  will  the  two  lines  x  +  2y  —  z  +  S  =  0,  Sx  —  y  -{- 
2z  +  l=0;  2  X— y  +  z— 2  =  0,  x  +  y  —  z  +  k  =  0  intersect  ? 

^  7.   Find  the  equation  of  the  plane  through  the  points  (1,  —  1,  2)  and 
(.3,  0,  1),  parallel  to  the  line  x  +  y  —  z  =  0,  2 x  +  y  +  z  =  0. 

oci        *i**i      1-         X— 2      V  +  I        z  i  X  —  3      w  +  4      z  +  2 

8.  Show  that  the  hues =  ^-^ —  = and  =  ^— J —  =  — ■ — 

3  3-2  _  1  3  2 

intersect,  and  find  the  equation  of  the  plane  determined  by  them. 

9.  Find  the  equation  of  the  plane  through  the  point  (a,  b,  c),  parallel  to 
each  of  the  lines,  "i^^  =  ^^^  =  l^lii ;  "^^^^  =  y^^^  =  ^-^^^. 

10.  Find  the  equation  of  the  plane  through  the  origin  and  perpendicular 
to  the  line  'i  x  —  y  +  i  z  -\-  ii  =  0,  x  +  y  —  z  =  0. 

11.  Find   the   value  of   k  for   which   the   lines  ^'  ~  '^  =  -^-^ —  = ^; 

2  k         k  +  l  5     ' 

X  —  I      V  +  5      z  +  2  ,.     , 

— - —  =  --^—^ —  = are  perpendicular. 

3  1  k-2         ^     ^ 

12.  Find  the  values  of  k  for  which  the  planes  kx  —  5  y  +  (k  +  S^z  +  3  =  0 
and  (k  —  l)x  +  ky  +  z  =  0  are  perpendicular. 

13.  Find  the  equations  of  the  line  through  the  point  (2,  3,  4)  which  meets 
the  I'-axis  at  right  angles. 

22.    Distance  from  a  point  to  a  line.     Given  the  line 

X  —  CCi  _  7/  —  Pi  _  Z  —  Zi 

X  fJL  V 

and  the  point  P^  =  (x^,  y^,  z^^  not  lying  on  it.     It  is  required  to  find 
the  distance  between  the  point  and  the  line. 


24 


PLANES  AND   LINES 


[Chap.  IL 


Fig.  13. 


Let  Pi  =  (.Ti,  y^,  z,)  (Fig.  13)  be  any 
point  on  the  line ;  let  P  be  the  foot  of 
the  perpendicular  from  Pj  o^  the  line ; 
0  the  angle  between  the  given  line  and 
•-j^  the  line  P1P2;  let  d  be  the  length  of 
the  segment  PiP^.     We  have  (Fig.  13) 

P,P2  =  P^P^^  sin^  0  =  (r~-  cP  cos^  e. 


The  direction  cosines  of  the  line  P^P^  are  '-- —,  — —, 

d  d 

from  which  (Art.  5) 


d 


d 


>j2  -  yi  _,_  ^.  ^2  -  zi 


d 


d 


Hence, 


d'  cos^  e  =  {X,  -  x,f  +  (//,  -  y,y  4-  (z.  -  z,y 


(20) 


23.    Distance  between    two  non -intersecting    lines.       Given    the 
two  lines 

X-  Xi  _  y  -  y,  ^  z  -  z,  ^^^^^  x  -  x.^  ^  y  -  ?/,  _  z  -  z^ 


Ai 


/*i 


Vl 


V2 


which  do  not  intersect.  It  is  required  to  find  the  shortest  dis- 
tance between  them.  Let  A,  ^,  v  be  the  direction  cosines  of  the 
line  on  which  the  distance  is  measured.  Since  this  line  is  per- 
pendicular to  each  of  the  given  lines,  we  have,  by  Art.  5, 
Equations  (4)  and  (5), 


AZ 


/* 


/i-lVo  —  V,/A2 


V 


±1 


A1M2  ~  1^1  A2      sin  6 

where  6  is  the  angle  between 
the  given  lines. 

The  length  d  of  the  required 
perpendicular  is   equal    to   the 
projection  on  the  common  per- 
pendicular of  the  segment  PP',  ^  ^'°-  ^^• 
and  is  equal  to  the  projection  of  the  broken  line  PMNP'  (Fig.  14). 


Arts.  23,  24]     SYSTEM   OF   PLANES   THROUGH  A   LINE       25 


or 


d  =  ± 


sin  6 


Hi  — 1/2     fJ-i     H-2 
Zi  —  e,     I'l      V2 

EXERCISES 

1.   Find  the  distance  from  the  oritiin  to  the  line 


1  (21) 


X-  1  _y-3_g-2 


2  4  1 

2.  Find  the  distance  from  (1,  1,  ])  to  x  +  y  +  z  =  0,  Sx  —  2y  +  4z  =  0. 

3.  Find  tlie  perpendicular  distance  from  the  point  (—  2,  1,  3)  to  the  line 
x  +  2y-z  +  'i-0,Hx  —  y  +  2z  +  l-0. 

4.  What  are  the  direction  cosines  of  the  line  through  the  origin  and  the 
point  of  intersection  of  the  lines  x  -\-  2 y  —  z  +  3  =  0,  ox  —  y  +  2z  +  l  =  0; 
2x  —  2y  +  3z  —  2=0,x-y-z  +  S  =  0. 

5.  Determine  the  distance  of  the  point  (1,  1,  1)  to  the  line  a;  =  0,  y  —  0 
and  the  direction  cosines  of  the  line  on  which  it  is  measured. 

6.  Find  the   distance    between   the   lines   -  =  ^  =  ~  ~      and  ^  ~   - 

2-2  1  4 

^y-3^z+l 
2  -  1  ■ 

7.  Find  the  equations  of  the  line  along  which  the  distance  in  Ex.  6 
is  measured. 

8.  Find  the  distance  between  the  lines  2  x  +  y  —  z  =  0,  x—  y  -\-2z  =  3 
and  x  +  2y  —  3z  —  'i,  2x  —  3?/  +  40  =  5. 

9.  Express   the   condition  that  the  lines  ^  ~  ^1  -  ^  ~  ^i  -  ^  ~  ^1  ^  x  -  Xj 

h  mi  Hi  h 

=  y^ZLVl  =  IjlLll  intersect. 

7712  «2 

24.    System  of  planes  through  a  line.     If 

ii  =  A^x  +  B^y  +  C^z  +  7),  =  0, 
L.,  =  A2X  +  Boy  +  C.JX  +  Z>2  =  0 

ane  the  equations  of  two  intersecting  planes,  the  equation  fcjLj  + 
A:2Z/2  =  0  is,  for  all  real  values  of  k^  and  ^•2,  the  equation  of  a  plane 
passing  through  the  line  Li  =  0,  Lo  =  0.  For,  Jc^L^  +  kjj^  =  0  is 
always  of  the  first  degree  with  real  coefficients,  and  is  therefore 
the  equation  of  a  plane  (Art.  10);  this  plane  passes  through  the 
line  ij  =  0,  7^2  =  0,  since  the  coordinates  of  every  point  on  the  line 
satisfy  Z-i  =  0  and  Xg  =  0  and  consequently  satisfy  the  equation 


26  PLANES  AND   LINES  [Chap.  H. 

\Li  +  Tx-.L.,  =  0.  Conversely,  the  equation  of  any  plane  passing 
through  the  line  can  be  expressed  in  the  form  l\Li  +  k.^Lo  =  0, 
since  k^  and  ^^,  can  be  so  chosen  that  the  plane  k^Li  +  A'2L2  =  0 
will  contain  any  point  in  space.  Since  any  plane  through  the 
given  line  is  determined  by  the  line  and  a  point  not  lying  on 
it,  the  theorem  follows. 

To  find  the  equations  of  the  plane  determined  by  the  line  L^  =  0, 
Z/2  =  0,  and  a  point  P^  not  lying  on  it,  let  the  coordinates  of  Pj  be 
(xi,  Pi,  Zi).  If  Pi  lies  in  the  plane  k^L^  +  koL.,  =  0,  its  coordinates 
must  satisfy  the  equation  of  the  plane;  thus 

k,(A,x,  +  B,>ji  +  C\z,+I),)  +  k,(A,Xi  +  B.^j,  +  C.,z^  +  A)  =  0. 
On  eliminating  A'^  and  k^  between  this  equation  and  k^L^  +  AvLj  =  0, 
we  obtain 
0  =  {Aa,  +  B.^j,  +  a^i  +  A)(^4^^•  +  B,y  +  C,z  +  A) 

-  {A,x^  +  5,^1  +  C^z^  +  A)  {A^  +  AV  +  C-Z  +  A), 

as  the  equation  of  the  plane  determined  by  the  line  A  =  0,  A  =  0, 
and  the  point  Pj. 

It  will  be  convenient  to  write  the  above  equation  in  the  abbre- 
viated form 

A(a:'i)A('«-')  —  A(-^i)  A(-t')  =  0- 

The  totality  of  planes  passing  through  a  line  is  called  a  pencil 
of  planes.  The  number  k^/'k^  which  determines  a  plane  of  the 
pencil  is  called  the  parameter  of  the  pencil. 

If,  in  the  ecjuation 

A'l  A  +  ^'2  A  =^  0, 

A'l  and  k.,  are  given  such  values  that  the  coefficient  of  x  is  equal  to 
zero,  the  corresponding  plane  is  perpendicular  to  the  plane  aj  =  0. 
Since  this  plane  contains  the  line,  it  intersects  the  plane  cc  =  0  in 
the  orthogonal  projection  of  the  line  Lj  =  0,  A  =  0.  Similarly, 
if  fci  and  k^  are  given  such  values  that  the  coefficient  of  y  is  equal 
to  zero,  the  corresponding  plane  is  perpendicular  to  the  plane  y  =  0 
and  will  cut  the  plane  y  =  0  in  the  projection  of  A  =  ^i  A  =  0  on 
that  plane  ;  if  the  coefficient  of  z  is  made  to  vanish,  the  plane  will 
contain  the  projection  of  the  given  line  upon  the  plane  z  =0.  The 
three  planes  of  the  system  k^L^  +  k.L^^  0  obtained  in  this  way 
are  called  the  three  projecting  planes  of  the  line  T/j  =  0,  ij  =  0  on 
the  coordinate  planes. 


Art.  24]         SYSTEM  OF   PLANES  THROUGH  A  LINE 


27 


Since  two  distinct  planes 
passing  through  a  line  are 
sufficient  to  determine  the 
line,  two  projecting  planes  of 
a  line  may  always  be  em- 
ployed to  define  the  line.  If 
the  line  is  not  parallel  to  the 
plane  z  =  0,  its  projecting 
planes  on  a;  =  0  and  y  =  0  are 
distinct  and  the  equations  of 
the  line  may  be  reduced  to  the  form  (Fig. 


Fig.  15 


15) 


X  =  mz  +  a,  y  =  nz  +  h. 


(22) 


If  the  line  is  parallel  to  2  =  0,  the  value  of  k  for  which  the  coeffi- 
cient of  X  is  made  to  vanish  will  also  reduce  the  coefficient  of  y  to 

zero,  so  that  the  projecting  planes  on 
x"  =  0  and  on  ?/  =  0  coincide.  This 
projecting  plane  z  =c  and  the  projec- 
ting plane  on  z  =  0  may  now  be  chosen 
to  define  the  line.  If  the  line  is  not 
"X  parallel  to  the  X-axis,  the  equations 
oi  the  line  may  be  reduced  to  (Fig.  16) 


X  =  py  -\-  c,   z  =  c. 


(23) 


Finally,  if  the  line  is  parallel  to  the  A"-axis,  its  equations  may  be 
reduced  to  (Fig.  17) 


y^b,  z^  c. 


(24) 


If  the  planes  L^  =  0,  L,  =  0  are  par- 
allel but  distinct,  so  that 

Ao     B.,     a     Do' 


A 


"O 


Y 


X 


Fig.  17. 


then  every   equation  of  the  form  k^L^  -|-  kj^^  =  0,  except  when 

k       A        H       C 

--2=— =— i=— ij  defines   a   plane    parallel    to   the    given   ones. 
k\     A-y     Bi     Cz 

\  Conversely,  the  equation  of  any  plane  parallel  to  the  given  ones 

\can  be  written  in  the  form  k^Li  +  koL.,  =  0  by  so  choosing  k^ :  k^ 


28  PLANES  AND   LINES  [Chap.  II. 

tliat  the  plane  will  pass  through  a  given  point.  In  this  case  the 
system  of  planes  k-^L^  +  A;2L2  =  0  is  called  a  pencil  of  parallel  planes. 
Two  equations 

A  =  A^x  +  Bi!j  +  C,z  +  A  =  0, 

Li^  A.^  +  B.2y -\-  G^z -\- Di  =  0 

will  represent  the  same  plane  when,  and  only  when,  the  coefficients 
Ax,  Bi,  Ci,  Di  are  respectively  proportional  to  A2,  B2,  C2,  D^;  thus, 
when 

A2    B2     C2    A' 

These  conditions  may  be  expressed  by  saying  that  every  deter- 
minant of  order  two  formed  by  any  square  array  in  the  system 

A,    A     Ci    D, 
A2    A     Q    D., 
shall  vanish. 

In  this  case  multipliers  ki,  k^  can  be  found  such  that  the  equa- 
tion k^Li  +  kJjo  =  0  is  identically  satisfied. 

Conversely,  if  multipliers  k^,  ko  can  be  found  such  that  the  pre- 
ceding identity  is  satisfied,  then  the  equations  Li  =  0,  L2  =  0 
define  the  same  plane. 

EXERCISES 

'^  1.  Write  the  equation  of  a  plane  through  the  line  7 x  +  2y  ~  z  —  S  =  0, 
3x~3y  +  2z  —  5  =  0  perpendicular  to  the  plane  2x-\-y  —  2z  =  0. 

2.  What  is  the  equation  of  the  plane  determined  by  the  line  2x  —  Sy  — 
z  +  2  =  0,  x-y  +  iz  =  S  and  the  point  (3,  2,  —  2)  ? 

/  3.   Determine    the    equation    of    the   plane   passing    through  the    line 

^  Q  y_L4  Z  7 

X  +  2  z  =  i,  y  —  z  =  8  and  parallel  to  the  line =  ^— !^ —  = . 

^  ^  112 

4.    Does  the  plane  x  +  2y  —  z  +  '4  =  0   have  more   than   one   point  in 

common  with  the  line  Sx  —  y  +  2z+l=0,  2x  — Sy  +  Sz-2  =  0? 

^5.    Determine  the  equations  of  the  line  through  (1,  2,  3)  intersecting  the 

two  lines  x  +  2  (/-3.j=0,  >/— 4,j  =  4  and  2x-y  +3^  =  3.  3x  +  y  -\-2z  +  1  =  0. 

25.  Application  in  descriptive  geometry.  A  line  may  be  repre- 
sented by  the  three  orthogonal  projections  of  a  segment  of  the  line, 
each  drawn  to  scale.  Consider  the  X>^-plane  (elevation,  or  verti- 
cal plane)  as  the  plane  of  the  paper,  and  the  XF-plane  as  turned 
about  the  ,Y-axis  until  it  coincides  with  the  XZ-plane.     The  pro- 


Arts.  25,  26] 


BUNDLES   OF   PLANES 


29 


^x 


jections  iu  the  XF-plaue  are  thus  drawn  to  scale  on  the  same 
paper  as  projections  on  the  XZ-plane,  but  points  are  distinguished 

by  different  symbols,  as  P',  P^.  q a  Z 

The  XF-plane  is  called  the  plan 
or  horizontal  plane.  Finally,  let 
the  FZ-plane  be  turned  about  the 
Z^axis  until  it  coincides  with  the 
XZ-plane,  and  let  figures  iu  the 
new  position  be  drawn  to  scale. 
This  is  called  the  end  or  profile 
plane.  Thus,  in  the  figure  (Fig. 
18),  a  segment  PQ,  wherein 
P=(7,  4,  8),  Q  =  (13,  9,  12), 
may  be  indicated  by  the  three  segments  P'Q',  PiQi,  PpQp- 

Example.    Find  the  equations  of  the  projecting  planes  of  the  line 

2x  +  32/  —  42  =  5,     x  —  iy  +  5z  =  6. 
Here,  Li  =  2  x  +  S  y  —  i  z  —  ^,     L2  =  x  —  iy  +  5z  —  6, 

kiLi  +  k2L2  =(2  ki  +  k2)x  +  (3  ki  -  4  A-^)?/ 

+  (_  4  A-i  +  5  hi)z  +  (  -  5  fci  -  6  A;2)  =  0. 

If  ki  =  —2  k\,  the  coefficient  of  x  disappears ;  thus  the  equation  of  the 
plane  projecting  the  given  line  on  the  plane  .r  =  0  is 
11  ?/-  142  +  7  =0. 

7.         q 

If  —  =:  -,  the  coefficient  of  y  vanishes;  the  projecting  plane  on  y  =  0  is 
ki     4 


found  to  be  1 1  X 


38. 


ko      4 
Finally,  if  -^  =  -,  the  projecting  plane  on  2=0  is  found. 
ki      b 


Its  equation 


is  14  X  —  ?/  =  49. 

EXERCISES 

Find  the  equations  of  the  projecting  planes  of  each  of  the  following  lines  : 
-'1.    z  +  2  y  -  3  2  =  4,         2  x  -  3  J/  +  4  2  =  5. 
2     2x  +y  +  z  =  0,  x  —  y  +  2  z  =  S. 

3.  X  +  t/  +  2  =  4,  X-  y  +  3z  =  4. 

4.  Au-  +  Bxy  +  Ciz  -f-  Z)i  =  0,     .4oX  +  B-.y  +  C\z  +  Z>2  =  0. 

26.    Bundles   of  planes.     The  plane  L^  =  A-^x  +  B^y  +  C^z  -\-  D^ 

=0  will  belong  to  the  pencil  determined  by  the  planes  A=  Oj  L2=0, 
assumed  distinct,  when  three  numbers  k\,  ko,  k^,  not  all  zero,  can 
be  found  such  that  the  equation  k^L^  +  k^Lo  ■+■  k^L^  =  0  is  identi- 


30  PpiNES  AND   LINES  [Chap.  II. 

cally  satisfied  for  all  values  of  x,  y,  z.  This  condition  requires  that 
the  four  equations  \Ay  +  hoA.,  +  ^s-^j  =  0,  k^B^  +  k^B^  +  Jc^B^  =  0, 
A^iCi  +  k-yC.  +  k^Cs  =  0,  k^Di  +  k^D.  +  ^^sA  =  0  are  satisfied  by 
three  numbers  k^,  k^,  k^,  not  all  zero ;  hence,  that  the  four  equa- 
tions 


I  A,B,G,  1=0,        I  B,C,D,  1  =  0,        1  C,D,A,  1=0,        \  D,A,B, 
are  all  satisfied,  wherein  we  have  written  for  brevity, 


0 


AiB^C^i 


Ay 

A 

c, 

A. 

B. 

c. 

A 

B, 

c. 

etc. 


These  simultaneous  conditions  may  be  expressed  by  saying 
that  every  determinant  of  order  three  formed  by  the  elements 
contained  in  any  square  array  in  the  system 

A,    B,    c;    D, 

A,    B,     C,    D, 

A3    B3     63     D3 
shall  vanish. 

Conversely,  if  these  conditions  are  satisfied,  then  three  con- 
stants ^'i,  A'2,  k^  can  be  found  such  that  the  equation  k^Li  +  kzL^ 
-\-  k^L^  =  0  is  identically  satisfied,  and  the  three  planes  L^  =  0, 
io  =  0,  -L3  =  0  belong  to  the  same  pencil. 


Let 


L,  =  A,x  +  B,y  +  C',2  +  D,  =  0, 
L.  =  A^x  +  B.ai  +  C.^  +  A  =  0, 
L,  =  A,x  +  B,v  +  C,z  +  X>3  =  0 


be  the  equations  of  three  planes  not  belonging  to  a  pencil.  If  we 
solve  these  three  equations  for  (x,  y,  z),  we  find  for  the  coordinates 
of  the  point  of  intersection  of  the  three  planes,  in  case  |  ^diCoCsl 

IAAC3I  Ul.ACsl  \A,B.D, 


A,B,C3 


AB^Cs 


AxBiCs 


(25) 


If  \AiB.2C\\  =0,  but  not  all  the  determinants  in  the  numerators 
of  (25)  are  zero,  no  set  of  values  of  x,  y,  z  will  satisfy  all  three  ' 
equations.     In  this  case,  the  line  of  intersection  of  any  two  of  the 
planes  is  parallel  to  the  third.     For,  if  L^  =  0  and  L,  =  0  intersect, 


•» 


Arts.  26,  27]  PLANE  COORDINATES  31 

the  direction  cosines  of  their  Hue  of  intersection  are  proportional 
(Art.  18)  to 

B,C^  -  BoC„         C,A,  -  aA„        A^B,  -  A.B,. 

The  condition  that  this  line  is  parallel  to  the  plane  Ls  =  0  is 
(Art.  21) 

A,{B,a  -  B,C,)  +  B,(C,A,  -  G,A,)  +  C,{A,B,  -  A,B,)  =  0, 

which  is  exactly  the  condition  |  A^B^C:^  j  =  0.  The  proof  for  the 
other  lines  and  planes  is  found  in  the  same  way. 

If  at  least  one  of  the  determinants  |  A^BoC^  \,  \  D^BoCs  \,  |  A^DoC^  \, 
and  I A1B2D3 1  is  not  zero,  the  system  of  planes 

A-jZ/i  +  Jc.Jj.,  +  k^L^  =  0 

is  called  a  bundle.  If  \ABC\^  0,  all  the  planes  of  the  bundle 
pass  through  the  point  (25),  since  the  coordinates  of  this  point 
satisfy  the  equation  of  every  plane  of  the  bundle.  Conversely, 
the  equation  of  every  plane  passing  through  the  point  (25)  can  be 
expressed  in  this  form.  This  point  is  called  the  vertex  of  the 
bundle.  If  \ABC\  =  0,  all  the  planes  of  the  bundle  are  parallel 
to  a  fixed  line  (such  as  L^  =  0,  L.,  =  0).  In  this  case,  the  bundle 
is  called  a  parallel  bundle. 

27.    Plane  coordinates.     The  equation  of  any  plane  not  passing 
through  the  origin  may  be  reduced  to  the  form 

ux  +  vy  -t-  wz  +  1=0.  (26) 

When  the  equation  is  in  this  form,  the  position  of  the  plane  is 
fixed  when  the  values  of  the  coefficients  «,  v,  w  (not  all  zero)  are 
known;  and  conversely,  if  the  position  of  the  plane  (not  passing 
through  the  origin)  is  known,  the  values  of  the  coefficients  are 
fixed.  Since  the  numbers  (a,  v,  iv)  determine  a  plane  definitely, 
just  as  (x,  y,  z)  determine  a  point,  we  shall  call  the  set  of  num- 
bers («,  V,  ic)  the  coordinates  of  the  plane  represented  by  equation 
(26).  Thus,  the  plane  (3,  5,  2)  will  be  understood  to  mean  the 
plane  whose  equation  is  3  a;  +  5  ?/  -f  2  2;  +  1  =  0.  Similarly,  the 
equation  of  the  plane  (2,  0,  —  1)  is  2  ic  —  2;  + 1  =  0. 

If  u,  V,  IV  are  different  from  zero,  they  are  the  negative  recipro- 
cals of  the  intercepts  of  the  plane  (u,  v,  w)  on  the  axes  (Art.  12). 


32  PLANES  AND  LINES  [Chap.  1L 

If  u  =  0,  the  plane  is  parallel  to  the  X-axis ;  if  u  =  0,  -y  =  0,  the 
plane  is  parallel  to  the  XF-plane.  The  vanishing  of  the  other 
coefficients  may  be  interpreted  in  a  similar  way. 

28,  Equation  of  a  point.  If  the  point  {x^,  y^,  z^  lies  in  the 
plane  (26),  the  equation 

ux^  +  vy^  +  icz^  +1  =  0  (27) 

must  be  satisfied.  If  x-^,  y^,  z^  are  considered  fixed  and  u,  v,  w 
variable,  (27)  is  the  condition  that  the  plane  (m,  v,  iv)  passes 
through  the  point  (a-„  y^,  Zi).  For  this  reason,  equation  (27)  is 
called  the  equation  of  the  point  (a-j,  yi,  Zi)  in  plane  coordinates. 

Thus,  u-5v  +  2iv-\-l  =0 

is  the  equation  of  the  point  (1,  —  5,  2)  ;  similarly, 

3u  +  IV  +  1  =0 

is  the  equation  of  the  point  (.3,  0,  1). 

If  equation  (27)  is  multiplied  by  any  constant  different  from 
zero,  the  locus  of  the  equation  is  unchanged.  Hence,  we  have 
the  following  theorem : 

Theorem.     TJie  linear  equation 

Au  +  Bv+Civ  +  D  =  0     (Z)  ^  0) 

is  the  equation  of  the  point  (—,   — ,        ]  in  plane  coordinates. 

Thus,   u  —  5v— 3^0  —  2  =  0    is     the    equation   of    the    point 
- 1     5     3^ 

2   '   2'   2^ 

The  condition  that  the  coordinates  (w,  v,  w)  of  a  plane  satisfy 
two  linear  equations 

uxi  +  vyi  -\-  u'Zi  4-1=0,         ^1X2  +  vy.,  +  wz.,  +1=0 

is  that  the  plane  passes  through  the  two  points  (x^,  y^,  z^  and 
{X2,  2/2)  ^-i)  and  therefore  through  the  line  joining  the  two  points. 
The  two  equations  are  called  the  equations  of  the  line  in  plane 
coordinates. 


r>2A^^JUCX^./s~ryJU   nru>4jt  tCirY\jL  .     S-  ,  *2  *% ,  1  tf 


Arts.  28,  29]         HOMOGENEOUS  COORDINATES  33 

EXERCISES 

1.  Plot  the  following  planes  and  write  their  equations  :  (1,  2,  i),  (3,  —  I, 

2.  Find  the  volume  of  the  tetrahedron  bounded  by  the  coordinate  planes 
and  the  plane  (—  h  ~  h  ~  i)- 

3.  What  are  the  coordinates  of  the  planes  whose  equations  are 

Tx  +  6y-^z+l=0,         x-6y  +  nz  +  o  =  0,         9.r-4=0? 

4.  Find   the   angle   which   the  plane   (2,   0,  5)    makes   with   the   plane 
(-1,  i2).  ^ 

5.  Write  the  equations  of  the  points  (1,  1,  1),  (2,  -  1,  ^,),  (6,  —2,  1). 

6.  What  are  the  coordinates  of  the  points  whose  equations  are 

2m-»-3w+1  =  0,         ?t +  2  to -3=0,         to -2  =  0? 
♦^7.    Find  the  direction  cosines  of  the  line 

3?<  —  •o4-2mj  +  1=0,         u  +  ^  V  +  2  w  -  I  =  0. 
8.    What  locus  is  determined  by  three  simultaneous  linear  equations  in 
(m,  V,  w)  ?  r^  ' 

*'9.  Write  the  equation  satisfied  by  the  coordinates  of  the  planes  whose 
distance  from  the  origin  is  2.  What  is  the  locus  of  a  plane  which  satisfies 
this  condition  ? 

29.    Homogeneous  coordinates  of  the  point  and  of  the  plane.     It  is 

sometimes  convenient  to  express  the  coordinates  x,  y,  z;  of  a  point 
in  terms  of  fonr  numbers  x',  y',  z' ,  t'  by  means  of  the  equations 

x'  v'  z' 

—  =  X,     ■^=y,     —  =  z. 

A  set  of  four  numbers  (x',  y',  z',  t'),  not  all  of  which  are  zero,  that 
satisfy  these  equations  are  said  to  be  the  homogeneous  coordinates 
of  a  point.  If  the  coordinates  (x',  y',  z',  t')  are  given,  the  point 
is  uniquely  determined  (for  the  case  t'  =  0,  compare  Art.  32), 
but  if  (x,  y,  z)  are  given,  only  the  ratios  of  the  homogeneous 
coordinates  are  determined,  since  (x',  y',  z',  t')  and  (kx',  ky',  kz',  kt') 
define  the  same  point,  k  being  an  arbitrary  constant,  different 
from  zero. 

Similarly,  if  the  coordinates  of  a  plane  are  (w,  v,  iv),  four  num- 
bers (?t',  v',  w',  s'),  not  all  of  which  are  zero,  may  be  found  such 

that 

u'  v'  w' 

—  =  u,      -  =  v,      —-10. 

s'  s  s' 


I^^n^z--^-^-  ^^o^ 


34  PLANES  AND    LINES  [Chap.  IL 

The  set  of  numbers  (?<',  v',  ?t'',  s')  are  called  the  homogeneous  coordi- 
nates of  the  plane. 

Where  no  ambiguity  arises,  the  accents  will  be  omitted  from 
the  homogeneous  coordinates. 

30.  Equation  of  a  plane  and  of  a  point  in  homogeneous  coordinates. 
If,  in  the  equation  If,  in  the  equation 

Ax  +  By-\-Cz-\-  D  =  0  An  +  By  +  Cjo  +  i)  =  0 
{D^O,  and  A,  B,  C  are  not  all  (D  ^  0,  and  A,  B,  C  are  not  all 
zero)  the  homogeneous  coordi-  zero)  the  homogeneous  coordi- 
nates of  a  point  are  substituted,  nates  of  a  plane  are  substituted, 
we  obtain,  after  multiplying  by  we  obtain,  after  multiplying  by 
t,  the  equation  of  the  plane  in  s,  the  eqiiation  of  the  point  in 
homogeneous  coordinates  homogeneous  coordinates 

Ax  +  Bt/  -\-Cz  +  Dt  =  0.  All  +  Bv  +  Civ  +  Ds  =  0. 

The   homogeneous    coordinates  The   homogeneous    coordinates 

of  this  plane  are  (^1,  B,  C,  D).  of  this  point  are  (A,  B,  C,  D). 

31.  Equation  of  the  origin.  Coordinates  of  planes  through  the 
origin.  The  necessary  and  sufficient  condition  that  the  plane 
whose  equation  is  ?<.r  +  ^n  -f-  vz  -\-  st  =  0  shall  pass  through  the 
origin  is  .s=0.  We  see  then  that  s  =  0  is  the  equation  of  the 
origin,  and  that  (u,  v,  iv,  0)  are  the  homogeneous  coordinates  of  a 
plane  through  the  origin.  Since  s  =  0,  it  follows  from  Art.  29  that 
the  non-homogeneous  coordinates  of  such  a  plane  cease  to  exist. 

32.  The  plane  at  infinity.  Let  {x,  y,  z,  t)  be  the  homogeneous 
coordinates  of  a  point.  If  we  assign  fixed  values  (not  all  zero) 
to  X.  y,  z  and  allow  t  to  vary,  the  corresponding  point  will  vary  in 
such  a  way  that,  as  ^  =  0,  one  or  more  of  the  non-homogeneous  co- 
ordinates of  the  point  increases  without  limit.  If  t  =  0,  the  non- 
homogeneous  coordinates  cease  to  exist,  but  it  is  assumed  that 
there  still  exists  a  corresponding  point  which  is  said  to  be  at 
infinity.  It  is  also  assumed  that  two  })oints  at  infinity  coincide 
if,  and  only  if,  their  homogeneous  coordinates  are  proportional. 

The  equation  of  the  locus  of  the  points  at  iniinity  is  ^  =  0. 
Since  this  equation  is  homogeneous  of  the  first  degree  in  x,  y,  z,  t, 
it  will  be  said  that  ^  =  0  is  the  equation  of  a  plane.  This  plane 
is  called  the  plane  at  infinity. 


Arts.  33,  34]  COORDINATE   TETRAHEDRON  35 

33.  Lines  at  infinity.  Any  finite  plane  is  said  to  intersect  the 
plane  at  infinity  in  a  line.  This  line  is  called,  the  infinitely  dis- 
tant line  in  the  plane.  The  equations  of  the  infinitely  distant  line 
in  the  plane  Ax  +  Bf/  +  Cz  +  Dt  =  0  are  Ax  +  By  +  Cz  =  0,t  =  0, 

Theorem.  Tlie  condition  that  two  finite  planes  are  ^mrallel  is 
that  they  intersect  the  plane  at  infinity  in  the  same  line. 

If  the  planes  are  parallel,  their  equations  may  be  written  in 
the  form  (Art.  15) 

Ax  +  By+  Cz  +Dt  =  0,     Ax  +  By+Cz  +  D't  =  0.         (28) 

It  follows  that  they  both  pass  through  the  line 

Ax->rBy+Cz=0,     t  =  0.  (29) 

Conversely,  the  equations  of  any  two  finite  planes  through  the 
line  (29)  may  be  written  in  the  form  (28).  The  planes  are  there- 
fore parallel. 

34.  Coordinate  tetrahedron.  The  four  planes  whose  equations 
in  point  coordinates  are 

a;  =  0,  y  =  0,  2  =  0,  t  =  0 
will  be  called  the  four  coordinate  planes  in  homogeneous  coordi- 
nates. Since  the  planes  do  not  all  pass  through  a  common  point, 
they  will  be  regarded  as  forming  a  tetrahedron,  called  the  coordi- 
nate tetrahedron.  The  coordinates  of  the  vertices  of  this  tetra- 
hedron are 

(0,0,0,1),  (0,0,1,0),  (0,1,0,0),  (1,0,0,0). 
The  coordinates  of  the  four  faces  in  plane  coordinates  are 

(0,0,0,1),  (0,0,1,0),  (0,1,0,0),  (1,0,0,0). 
The  equations  of  the  vertices  are  u  =  0,  v  =  0,  iv  =  0,  .s  =  0. 

EXERCISES 

1.  Find  the  iKm-homogeneous  coordinates  of  the  following  points  and 
plane.s :  ^ 

(h)     10  a; -3y-(- 15  =  0,  (e)    u  +  v-w-l=0, 

(c)    x-2  =  0,  (f)  2w+  11  =0. 

>^  2.    Determine  the  coordinates  of  the  infinitely  distant  point  on  the  line 
Sx -j-2  >j +  [,t  =  U,     2x—  \{)z  +  At  =  0. 


36 


PLANES  AND   LINES 


[Chap.  IL 


r3.  Show  that  if  Li{u)  =AiU  +  Biv  +  CiW  +  Dis  =  0,  and  Z2(m)  =^2« 
+  iJo??  +  C2W  +  D2S  =  0  are  the  equations  of  two  points,  the  equation  of  any 
point  on  the  joining  line  may  be  written  in  the  form  kiLi  +  k^Li  =  0. 
t-  ''4.  Show  that  the  planes  X  +  22/  +  72  —  3«=:0,  x  +  3?/+62  =  0,  x  +  4j/ 
+  52  —  2^  =  0  determine  a  parallel  bundle.  Find  the  equation  of  the  plane 
of  the  bundle  through  the  points  (2,  —  1,  1,  1),  (2,  5,  0,  1). 

35.    System  of  four  planes.     The  condition  that  four  given  planes 

L,  =  A,x  +  B^y  +  C,z  +  D,t  =% 
L2  =  A.2X  +  B^y  +  C.2Z  +  Dot  =  0, 
A  =  A,x  +  B,y  +  Qz  +  Dit  =  0, 
L,  =  A,x  +  B,y  +  dz  -{-D,t  =  0 

all  pass  through  a  point\is  that  four  numbers  {x,  y,  z,  t),  not  alfl 
zero,  exist  which  satisfy  the  four  simultaneous  equations.  The 
condition  is,  consequently,  that  the  determinant 

A,  B,  a  ^1 

Ao   B.   a   A 


A, 


B, 


A 
A 


is  equal  to  zero.  If  this  condition  is  not  satisfied,  the  four  planes 
are  said  to  be  independent.  When  the  given  planes  are  independ- 
ent, four  numbers  A,,  k,,  k^,  k^  can  always  be  found  such  that  the 
equation 

A'lLi  +  A'oLa  +  A-jLj  +  kjj^  =  0 

shall  represent  any  given  plane.  For,  let  ax  -j-  by  -\-  cz  +  d  —  0  be 
the  equation  of  the  given  plane.  The  two  equations  will  repre- 
sent the  same  plane  if  their  coefficients  are  proportional,  that  is, 
if  numbers  Atj,  k,,  k^,  k^,  not  all  zero,  can  be  found  such  that 

a  =  k^Ai  +  A;,  A  +  hA  +  A-4.14, 
b  =  k,B,  +  k,B2  +  k,B,  +  k,B„ 
c  =  A'lCi  -|-  kiCi  -f  ksC^  -H  ^4^4, 
d  =  k,D^  +  k,D,  +  k,D,  +  Jc^D,. 

Since  the  planes  are  independent,  the  determinant  of  the  coeffi- 
cients in  the  second  members  of  these  equations  is  not  zero,  and 
the  numbers  A;,,  k^,  k^,  ki  can  always  be  determined  so  as  to  satisfy 
these  equations. 


Art.  35] 


SYSTEM  OF  FOUR  PLANES 


37 


These  results,  together  with  those  of  Arts.  24,  26,  may  be  ex- 
pressed as  follows :  The  necessary  and  sufficient  condition  that  a 
system  of  planes  have  no  point  in  common  is  that  the  matrix* 
formed  by  their  coefficients  is  of  rank  four ;  the  planes  belong  to 
a  bundle  when  the  matrix  is  of  rank  three ;  the  planes  belong  to 
a  pencil  when  the  matrix  is  of  rank  two ;  finally,  the  planes  all 
coincide  when  the  matrix  is  of  rank  one.  We  shall  use  the  ex- 
pression "  rank  of  the  system  of  planes  "  to  mean  the  rank  of  the 
matrix  of  coefficients  in  the  equations  of  the  planes. 

\  \r  EXERCISES 


^' 


\jj^     1.   Determine  the  nature  of  the  following  systems  of  planes  : 
'       l^a)  2x  —  5y  +  z  —  3t  =  0,  x  +  y  + 'iz  —  5t  =0,  x  +  Sy  +  6z-t  =  0. 
(ft)   3x  +  iy  +  5z-5t  =  0,    6x  +  5y  +  9z-l0t  =  0,   3x  +  Sy  +  5z 
-5«  =  0,  x—y  +  2z  =  0. 

(c)  2x-f4j/  =  0,  Hx  +  ly  +  2z  =  0,  Sx  +  iy  -  2z  +  ?,t  =  0,  x  =  0. 

(d)  2x  +  £>y  +  Sz-0,  7y-[,z  +  it  =  0,  x-y  +  iz  =  8t. 

JL^^  ^-^.   Show  that  the  line  x  +  ?jy  —  z  +  t  =  0,  2x-y  +  2z  —  St-0  lies  in 
the  plane  7  x  +  1  y  +  z  —  3t  =  0. 
^3.   Determine  the  conditions  that  the  planes 

X  =  cy  +  bz,   y  =  ax  +  cz,   z  =  bx  +  ay 
shall  have  just  one  common  point ;  a  common  line  ;  are  identical. 

4.  Prove    that    the    planes   2x  —  Sy  —  7z  =  0,    3  x  —  14  y  —  13  z  =  0, 
8x  —  31?/  —  33  2  =  0  have  a  line  in  common,  and  find  its  direction  cosines. 

5.  Show  that  the  planes  3x  —  2y  —  t  =  0,   ix  —  2z  —  2  t  =  0,   4x  -\-  4y 
—  b z  =0  belong  to  a  parallel  bundle. 


*  Any  rectangular  array  of  uumbers 

Ai     2?i     C'l     Di     ...     3/] 
A.2    B.2     C'2     Di     -    Mi 


An      B„      Cn      Z)„ 


Mn 


is  called  a  matrix.  Associated  with  every  matrix  are  other  matrices  obtained 
by  suppressing  one  or  more  of  the  rows  or  one  or  more  of  the  columns  of  the 
given  matrix,  or  both  ;  in  particular,  associated  with  every  square  matrix,  that 
is,  one  in  which  the  number  of  rows  is  equal  to  the  number  of  columns,  is  a  de- 
terminant whose  elements  are  the  elements  of  the  matrix.  Conversely,  associated 
with  every  determinant  is  a  square  matrix,  formed  by  its  elements.  We  shall 
use  the  word  rank  to  define  the  order  of  the  non-vanishing  determinant  of  high- 
est order  contained  in  any  given  matrix.  The  rank  of  tlie  determinant  is  defined 
as  the  rank  of  the  matrix  formed  by  the  elements  of  the  determinant. 


CHAPTER   III 


TRANSFORMATION  OF   COORDINATES 

The  coordinates  of  a  point,  referred  to  two  different  systems 
of  axes,  are  connected  by  certain  relations  which  will  now  be 
determined.  The  process  of  changing  from  one  system  of  axes 
to  another  is  called  a  transformation  of  coordinates. 

36.  Translation.  Let  the  coordinates  of  a  point  P  with  respect 
to  a  set  of  rectangular  axes  OX,  OY,  OZ  be  (.c,  y,  z)  and  with 
respect  to  a  set  of  axes  O'X',  0'  Y',  0' Z',  parallel  respectively 
to  the  first  set,  be  (.«',  y',  z').  If  the  coordinates  of  0',  referred 
to  the  axes  OX,  0  Y,  OZ  are  {h,  k,  l)  we  have  (Fig.  19) 

x  =  x'  +  h,    y  =  y'-\-J,;    z  =  z'  +  l.  (1) 

For,  the  projection  on  OX  of  OP  is  equal  to  the  sum  of  the  pro- 
jections of  00'  and  O'P  (Art.  2),  but  the  projection  of  OP  is  x, 

of  00'  is  h,  and  of  O'P  is  x';  hence 
x  —  x'  +  h.  The  other  formulas  are 
derived  in  a  similar  way.  Since 
the  new  axes  can  be  obtained  from 
the  old  ones  by  moving  the  three 
coordinate  planes  parallel  to  the 
X-axis  a  distance  h,  then  parallel 
to  the  y-axis  a  distance  k,  and 
parallel  to  the  .Z-axis  a  distance 
I,  without  changing  their  directions,  the  transformation  (1)  is 
called  a  translation  of  axes. 

37.  Rotation.  Let  the  coordinates  of  a  point  P,  referred  to  a 
set  of  rectangular  axes  OX,  OY,  OZ,  be  x,  y,  z,  and  referred  to 
another  rectangular  system  OX',  OY',  OZ'  having  the  same  origin, 
be  x',  y',  z'.  Let  x'  =  OL',  y'  =  L'M',  z'  =  M'PiFig.  20);  and  let 
the  direction  cosines  of  OX',  referred  to  OX,  O  Y,  OZ,  be  A,,  fxi,  vi ; 
those  of  OY'  be  Aj,  fi.2,  vo,  and  of  OZ'  be  A3,  yu.3,  v^. 

38 


Art.  37] 


ROTATION 


39 


We  shall  show  that 

X  =  Aix'  +  XoXi'  +  Ajz', 

y  =  ii.,x'  +  /xo?/'  +  M32',  (2) 

2  =  vix'  +  v^y'  +  v^z'. 

For,  the  projection  of  OP  (Fig.  20)  on  the  axis  OX  is  x.     The  sum 
of   the  projections  of   OL', 
L'M',  and  M'P  is  A,.r'  +  A.^^' 
+  A32:'. 

That  these  two  expres- 
sions are  equal  follows  from 
Art.  2.  The  second  and 
third  equations  are  obtained 
in  a  similar  way. 

The  direction  cosines  of 
OX,  0  Y,  and  OZ,  with  re- 
spect to  the  axes  OX',  OY', 
OZ'  are  Ai,  A,,  A3;  /ii,  /xo,  fi^; 
vi)  V2,  V3,  respectively.    If  Ave  '^"'  "  ' 

project  OP  and  0L=  x,  LM  =  y,  and  MP  =  z  on  OX',  OY',  and 
OZ',  we  obtain 


(2') 


x'  =  X^x  +  [x{y  +  v^z, 
y'  =  X^x  +  fi.y  +  v^z, 
z'  —  X^x  +  fji^y  +  v^z. 

The  systems  of  equations  (2)  and  (2')  are  expressed  in  con 
venient  form  by  means  of  the  accompanying  diagram. 


x< 

y' 

z' 

x 

K 

A3 

A3 

y 

H-i 

H-2 

H-3 

z 

Vl 

v. 

V3 

Since  Aj,  /aj,  v^  ;  A2,  /u,o,  v., ;  A3,  //.j,  V3    are  the  direction  cosines  of 
three  mutually  perpendicular  lines,  we  have  the  six  relations 

K^  +  f^l^  +  n^  =  1>  A1A2  +  /X1/U2  -f  ViVo  =  0, 

V  -f-  /tA2^  4-  V2^  =  1,  A2A3  +  /X21U.3  +  V2V3  =  0,  (3) 

K^  +  H-3'  +  »'3^  =  Ij  A3A1  +  /i,3/Ai  -f  J/3V1  =  0. 


40  TRANSFORMATION   OF   COORDINATES        [Chap.  III. 

We  have  seen  that  Ai,  A2,  A3;  /xj,  ^u,.,,  /^a^  v,,  vo,  v^  are  also  the  di- 
rection cosines  of  three  mutually  perpendicular  lines.  It  follows 
that 

^1^  +  A.2^  +  A-3^  =  Ij  KH-I  +  ^^2/^2  +  ^3/^3  =  0, 

P-i^  +  /A2^+  M3^  =  Ij  /AlVi   +  /A2l'2  +  H-3V3  =  0,  (4) 

Vl^  +  V2^  +  V3^  =  1,  ViAj    +  V2A2   +  V3A3  =  0. 

It  will  next  be  shown  that 

Aj  =  € (1X2V3  —  V2IJ.3),    A2  =  e (/X3V1  —  V3/A1),    A3  =  e  (/AiV2  —  V1/A2), 

/xj  =  e(v2A3  —  A2V3),    |Li2  =  e(v3Ai  —  A3V1),     ^3  =  e(viA2  —  AiVj),       (5) 

vj  =  e  (A2M3—  fJ.2>^3),    Vo  =  e  (Ag^i  —  ^3 Aj),    V3  =  e  (Aj/Xj  —  /XjAs), 

where  e=  ±1.     From  the  first  and  third  equations  of  the  last 
column  of  (4)  we  obtain 

Ai        ^        A2        ^         A3 

fUV3  —  V2/A3         /A3V1  —  V3/I.1         /XiVo  —   I'lfJ.o 

If  we  denote  the  value  of  these  fractions  by  e,  solve  for  Ai,  A2,  and 
A3  and  substitute  in  the  first  of  equations  (4),  we  obtain 

f^[(/tA2"3  —  ^2^3)-  +  (/A3V,  —  V3/A1)'  +(/tiV2  —  Viflof]  =  1. 

Since  the  lines  OY'  and  OZ'  are  perpendicular,  the  coefficient  of 
c2  is  unity  (Art.  5,  Eq.  (5)).    It  follows  that  e^  =  1  or  e  =  ±  1.    The 
first  three  of  equations  (5)  are  consequently  true.    The  other  equa- 
tions may  be  verified  in  a  similar  way. 
It  can  now  be  shown  that 


Ai  A2  A3 
fJ-i  P-2  ^3 

V,    V2   V3 


=  ±1.  (6) 


For,  expand  the  determinant  by  minors  of  the  elements  of  the 
first  row,  and  substitute  for  the  cofactors  of  Aj,  A2)  A3  their  values 
from  (5),     The  value  of  the  determinant  reduces  to 

It  will  be  shown  in  the  next  Article  that  if  e  =  1,  the  system  of 
axes  0-X'Y'Z'  can  be  obtained  by  rotation- from  0-XYZ.  If 
c  =  —  1,  a  rotation  and  reflection  are  necessary. 


Art.  38]         ROTATION  AND   REFLECTION   OF  AXES 


41 


38.  Rotation  and  reflection  of  axes.  Having  given  three  mutu- 
ally perpendicular  directed  lines,  forming  the  trihedral  angle 
0-XYZ (Fig.  21),  and  three  other  mutually  perpendicular  directed 
lines  through  0,  forming  the  trihedral  angle  0-X'Y'Z',  we  shall 
show  that  the  trihedral  angle  0-XYZ  can  be  revolved  in  such 
a  way  that  OX  and  OZ  coincide  in  direction  with  OX'  and  OZ', 
respectively.  OY  will  then  coincide  with  OY'  or  will  be  di- 
rected oppositely  to  it. 

Let  Xy  be  the  line  of  intersection  of  the  planes  XOY  and 
X'OY'.     Denote  the  angle  ZOZ'  by  6,  the  angle  XOX  by  </>,  and 


Z'  k  Z 


the  angle  XOX'  by  ij/.  Let  the 
axes  0-XYZ  be  revolved  as  a 
rigid  body  about  OZ  through  the 
angle  <f>,  so  that  OX  is  revolved 
into  the  position  OX.  Denote  the 
new  position  of  OF  by  OY^,  so 
that  the  angle  YOY^  =  (f>.  The 
trihedral  angle  0-XYZ  is  thus  re- 
volved into  0-X^Y,Z.  Now  let 
0-XY^Z  be  revolved  about  OX 
thi-ough  an  angle  6,  so  that  OZ 
takes  a  position   OZ',  and   OFj,  a  F^^-  ^l- 

position  OY2.  Then  the  angle  ZOZ' =  angle  YiOY2  =  e.  The 
trihedral  angle  O-X^Y^Z  is  thus  brought  into  the  position 
O-XY2Z'.  Finally,  let  the  trihedral  angle  in  this  last  position 
be  revolved  about  OZ'  through  an  angle  ij/,  so  that  O^is  revolved 
into  OX'.  By  the  same  operation  OFis  revolved  into  a  direction 
through  0  perpendicular  to  OX'  and  to  OZ'.  It  either  coincides 
with  OY'  or  is  oppositely  directed.  In  the  first  case  the  trihedral 
0-XYZ  has  been  rotated  into  the  trihedral  0-X'Y'Z'.  In  the 
second  case  the  rotation  must  be  followed  by  changing  the  direc- 
tion of  the  F-axis.  This  latter  operation  is  called  reflection  on  the 
plane  i/  =0.     It  cannot  be  accomplished  by  means  of  rotations. 

In  case  the  trihedral  0-XYZ  can  be  rotated  into  0-X'Y'Z', 
the  number  «  (Art.  37)  is  positive  ;  otherwise,  it  is  negative.  For, 
during  a  continuous  rotation  of  the  axes,  the  value  of  e  (Eq.  (6)) 
cannot  change  discontinuously.  If,  after  the  rotation,  the  trihe- 
drals  coincide,  we  have,  in  that  position,  Ai  =  /Hj  =  V3  =  1  and  the 


42  TRANSFORMATION   OP   COORDINATES       [Chap.  III. 

other  cosines  are  zero,  so  that  (Eq.  (G))  e  =  1*.  If,  however,  at  the 
end  of  the  rotation,  0  Y  and  0  Y'  are  oppositely  directed,  Aj  = 
V3  =  1,  /A2  =  —  1>  and  e  =  —  1. 

39.  Euler's  formulas  for  rotation  of  axes.  Let  the  coordinates  of 
a  point  P  referred  to  0-XYZ  be  {x,  y,  z),  referred  to  O-NY^Z  be 
(xi,  2/1,  ^i),  referred  to  O-NY2Z  be  ix2,  y-i,  z^),  and  referred  to 
0-X'  Y'Z'  be  {x',  y',  z'),  (Fig.  21). 

In  the  first  rotation,  through  the  angle  ^,  z  remains  fixed. 
Hence,  from  plane  analytic  geometry, 

z  =  2,,     X  =  Xi  cos  <i>  —  yi  sin  <^,     y  =  x^  sin  <^  +  2/1  cos  <p. 

In  the  rotation  through  the  angle  6,  x^  remains  fixed.  Hence 
we  have 

Xi  =  X2,     2/1  =  1/2  cos  0  —  Zo  sin  0,     z^  =  ?/2  sin  Q  -\-  z^  cos  ^. 

Finally,  if  O-X' Y' Z'  can  be  obtained  from  0-XYZ  by  rotation, 
22  remains  fixed,  and  we  have 

z^  =  z',     X.,  =  x'  cos  ijy  —  y'  sin  i//,     y^  =  x'  sin  ^  -\-  y'  cos  i/'. 

On  eliminating  x.^,  y2,  z^;    x^,  y^,  z^,  the  final    result  is  obtained, 

namely : 

X  =  x'  (cos  (fi  COS  i/'  —  sin  <^  sin  \p  cos  0)  —  ?/'(cos  <^  sin  ij/ 

+  sin  <)!>  cos  ij/  cos  ^)  +  z'  sin  </>  sin  ^. 

y  =  x'  (sin  <^  cos  i/'  +  cos  <^  sin  ip  cos  0)—  ^(sin  ^.sin  \f/ 

—  cos  </>  cos  li'  cos  6)  —  z'  cos  <f>  sin  ^. 

z  =  x'  sin  i/^  sin  6  +  y'  cos  i//  sin  ^  +  z'  cos  ^. 

If  0-X' Y'Z'  cannot  be  obtained  from  0-XYZ  hy  rotation,  the 
sign  of  y'  should  be  changed.  These  formulas  are  known  as 
Euler's  formulas. 

4-0.  Degree  of  an  equation  unchanged  by  transformation  of  co- 
ordinates. If  in  an  equation  F{x,  y,  z)  =  0  the  values  of  x,  y,  z  are 
replaced  by  their  values  in  any  transformation  of  axes  the  degree 
of  F  cannot  be  made  larger,  since  x,  y,  z  are  replaced  by  linear  ex- 
pressions in  x',  y',  z'.  But  the  degree  of  the  equation  cannot  be 
made  smaller,  since  by  returning  to  the  original  axes  and  to  the 
original  equation,  it  would  be  made  larger,  which  was  just  seen  to 
be  impossible. 


Art.  40]  EXERCISES  43 


EXERCISES 

1.  Transform  the  equation  x'^  —  3  yz  +  y-  —  6  x  +  z  =  0  to  parallel  axes 
through  the  point  (1,  —1,2), 

2.  By  means  of  equations  (2)  show  that  the  expression  x^  +  y^  -\-  z-  is  un- 
changed by  rotation  of  the  axes.     Interpret  geometrically. 

3.  Show  that  the  lines  x  =  ^  =  ^;    -  =  -^  =  z  ;    -  =y  =  -^—  are  mu- 

4      22-1  '2^-3 

tually  perpendicular.  "Write  the  equations  of  a  transformation  of  coordinates 
to  these  lines  as  axes. 

*^4.  Translate  the  axes  in  such  a  way  as  to  remove  the  first  degree  terms 
from  the  equation  x"^  -2y^  +  Qz^  -  \Gx  — 4y  —  24:Z  +  37  =  0. 

^.  Show  that  the  equation  ax  +  by  +  cz  +  s  =  0  may  be  reduced  to  x  =  0 
by  a  transformation  of  coordinates. 

^'  6.  Find  the  equation  of  the  locus  11  x-  +  10  y-  +  6  z'^  —  8  yz  +  i  zx  —  12  xy 
—  12  =  0  when  lines  through  the  origin  whose  direction  cosines  are  ^,  |,  |  ; 
h  h  ~  ^>  ~  h  h  ~  i  ^^®  taken  as  new  coordinate  axes. 

7.  Show  that  if  0-X'  Y'Z'  can  be  obtained  from  0-XYZ  by  rotation,  and 
if  OY  can  be  made  to  coincide  with  OX  by  a  revolution  of  90  degrees, 
counterclockwise,  as  viewed  from  the  positive  end  of  the  /^-axis,  then  OY' 
can  be  revolved  into  OX'  by  rotating  counterclockwise  through  90  degrees  as 
viewed  from  the  positive  Z'-axis. 

8.  Derive  from  Ex.  7  a  necessary  and  sufficient  condition  that  0-X  Y'Z 
can  be  obtained  from  0-X  YZ  by  rotation. 


CHAPTER   IV 

TYPES  OF  SURFACES 

41.  Imaginary  points,  lines,  and  planes.  In  solving  problems 
that  arise  in  analytic  geometry,  it  frequently  happens  that  the 
values  of  some  of  the  quantities  x,  y,  z  which  satisfy  the  given 
conditions  are  imaginary.  Although  we  shall  not  be  able  to  plot 
a  point  in  the  sense  of  Art.  1,  when  some  or  all  of  its  coordinates 
are  imaginary,  it  will  nevertheless  be  convenient  to  refer  to  any 
triad  of  numbers  x,  y,  z,  real  or  imaginary,  as  the  coordinates  of  a 
point.  If  all  the  coordinates  are  real,  the  point  is  real  and  is  de- 
termined by  its  coordinates  as  in  Art.  1 ;  if  some  or  all  of  the 
coordinates  are  imaginary  or  complex,  the  point  will  be  said  to  be 
imaginary.  Similarly,  a  set  of  plane  coordinates  u,  v,  w  will  de- 
fine a  real  plane  if  all  the  coordinates  are  real ;  if  some  or  all 
of  the  coordinates  are  imaginary,  the  plane  will  be  said  to  be 
imaginary. 

A  linear  equation  in  x,  y,  z,  with  coefficients  real  or  imaginary, 
will  be  said  to  define  a  plane,  and  a  linear  equation  in  ti,  v,  w, 
with  coefficients  real  or  imaginary,  will  be  said  to  define  a  point. 

The  equations  of  any  two  distinct  planes,  considered  as  simul- 
taneous, will  be  said  to  define  a  line.  It  follows  that  if  (.x-j,  y^,  z{) 
and  (.x-2,  y-,,,  2.>)  are  any  two  points  on  the  line,  then  the  coordinates 
of  any  othei"  point  on  the  line  can  be  written  in  the  form 
A-jXi  +  A-jXg,  etc.  The  line  is  also  determined  by  the  equations  of 
any  two  distinct  points  on  it. 

The  line  joining  two  imaginary  points  is  real  if  it  also  contains 
two  real  points.  If  P  =(a  -\-  ik,  b  -f  il,  c  +  im)  is  an  imaginary 
point,  the  point  P'  =(a  —  ik,  b  —  il,  c  —  im),  whose  coordinates 
are  the  respective  conjugates  of  those  of  P,  is  called  the  point 
conjugate  to  P.  The  line  joining  any  two  conjugate  points  is 
real ;  tlius  the  equations  of  the  line  PP'  are  Ix  —  ky  -{-  bk  —  al  =  0, 
(bm  —  d)x  +{ck  —  am)y  +(al  —  bk)z  =  0.     The  line  of  intersec- 

44 


Art.  41]       IMAGINARY   POINTS,   LINES,   AND   PLANES      45 

tion  of  two  imaginary  planes  is  real  if  through  it  pass  two  distinct 
real  planes.  The  line  of  intersection  of  two  conjugate  planes 
is  rer' 

Fr^ii^  the  preceding  it  follows  that  no  imaginary  line  can  con- 
tain more  than  one  real  point,  and  through  an   imaginary  line 
cannot  pass  more  than  one  real  plane.     If  a  plane  passes  through^ 
an  imaginary  point  and  not  through  its  conjugate,  the  plane  is  Uttc^ 
imaginary.      If  a  point  lies  in   an   imaginary   plane  and  not  in  j 
its  conjugate,  the  point  is  imaginary-. 

One  advantage  of  using  the  form  of  statement  suggested  in  this 
Article  is  that  many  theorems  may  be  stated  in  more  general  form 
than  would  otherwise  be  possible.  We  may  say,  for  example, 
that  every  line  has  two  (distinct  or  coincident)  points  in  common 
with  any  given  sphere. 

With  these  assiimptions  the  preceding  formulas  will  be  applied 
to  imaginary  elements  as  well  as  to  real  ones.  No  attempt  will  be 
made  to  give  to  such  f(n-mulas  a  geometric  meaning  when  imagi- 
nary quantities  are  involved. 

In  the  following  chapters,  in  all  discussions  in  which  it  is 
necessary  to  distinguish  between  real  and  imaginary  quantities, 
it  will  be  assumed,  unless  the  contrary  is  stated,  that  given  points, 
lines,  and  planes,  and  the  coefficients  in  the  equations  of  given 
surfaces,  are  real. 

'       '  EXERCISES 

1.    Show  that  the  point  (2  +  i,  1  +  3  i,  i)  lies  on  the  plane  x  —  2y  +  5  2=0. 

V2.  Find  the  coordinates  of  the  points  of  intersection  of  the  line  whose 
parametric  equations  are  (Art.  20)  x  =  1  +  ^^  d,  y  =  —  2  +  ^^  d,  z  =  5  —  \f  d, 
with  the  sphere  x'^  +  y'^  +  z^  =  1. 

'      ^3.    Show  that  the  line  of  intersection  of  the  planes  x  +  ?^  =  0,   (1  -j^  *)^^    a  • 

4.  Find  the  coordinates  of  the  point  of  intersection  of  the  line  through 
(3,  2,  -  2)  and  (4,  0,  3)  with  the  plane  x  +  3  y  +  (1  -  2  1)2^  +  1  =  0. 

'     5.    Find  the  equation  of  the  plane  determined  by  the  points  (5  -|-  i,  2,-2 

-  0,  (4  +  2  f,  -1  +  2  i,  0),  (i,  1  +  2  (\  1+3  0. 

6.  Determine  the  points  in  which  the  sphere  (x  -  l)'^  +  r/^  +  (0  +2)2  =  1 
intersects  the  X-axis. 


46  TYPES  OF  SURFACES  [Chap.  IV. 

42.  Loci  of  equations.  The  loons  defined  by  a  single  eqnation 
among  the  variables  x,  y,  z  is  called  a  surface.  A  point 
P=  (.x'l,  yi,  2i)  lies  on  the  surface  i^=  0  if,  and  only  if,  the  coor- 
dinates of  P  satisfy  the  equation  of  the  surface.  We  have  seen, 
for  example,  that  the  locus  of  a  linear  equation  is  a  plane.  More- 
over, the  locus  of  the  equation 

a;2  +  ^2  ^_  ^2  ^  1 

is  a  sphere  of  radius  unity  with  center  at  the  origin. 

The  locus  of  the  real  points  on  a  surface  may  be  composed  of 
curves  and  points,  or  there  may  be  no  real  points  on  the  surface ; 
for  example,  the  locus  of  the  real  points  on  the  surface 

X^  +  7/2  =  0 

is  the  Z-axis ;  the  locus  of  real  points  on  the  surface 

•«'  +  y2  +  2^  =  0 

is  the  origin;  the  surface 

x'-\-if  +  z^  +  l  =  0 
has  no  real  points. 

If  the  equation  of  a  surface  is  multiplied  by  a  constant  different 
from  zero,  the  resulting  equation  defines  the  same  surface  as  be- 
fore; for,  if  P=  0  is  the  equation  of  the  surface  and  k  a  constant 
different  from  zero,  the  coordinates  of  a  point  P  will  satisfy  the 
equation  kF  =  0  if,  and  only  if,  they  also  satisfy  the  equation  F=0. 

The  locus  of  two  simultaneous  equations  is  the  totality  of  the 
points  whose  coordinates  satisfy  both  equations.  If  F{x,  y,  z)=0, 
fix,  y,  z)  =0  are  the  equations  of  two  surfaces,  then  the  locus  of 
the  simultaneous  equations  ^=0, /=0  is  the  curve  or  curves  in 
which  these  surfaces  intersect.  Every  point  on  the  curve  of  in- 
tersection may  be  imaginary. 

The  locus  of  three  simultaneous  equations  is  the  totality  of  the 
points  whose  coordinates  satisfy  the  three  simultaneous  equations. 


/?■ 


EXERCISES 

1.  Find  the  equation  of  the  locus  of  a  point  whose  distance  from  the  Z-axis 
is  twice  its  distance  from  the  JTF-plane. 

2.  Discuss  the  locus  defined  by  the  equation  x^  +  x'^  —  yf-. 

"^  3.   Find  the  equation  of  the  locus  of  a  point  the  sum  of  the  squares  of 
whose  distances  from  the  points  (1,  3,  —  2),  (6,  —  4,  2)  is  10. 


Arts.  43,  44]  PROJECTING  CYLINDERS  47 

r  4.   Find  the  equation  of  the  locus  of  a  point  whicli  is  three  times  as  far 
from  the  point  (2,  6,  3)  as  from  the  point  (4,  —  2,  4). 

5.  Find  the  equations  of  the  locus  of  a  point  wliich  is  5  units  from  the 
XF-plane  and  3  units  from  the  point  (3,  7,  1). 

'6.    Find  tlie  equations  of  tlie  locus  of  a  point  which  is  equidistant  from  the 
points  (2,  3,  7),  (3,  -4,  6),  (4,  3,  -2). 

7.  Find  the  coordinates  of  the  points  in  which  the  line  x  =  —  i,  z  =  2  in- 
tersects the  cylinder  y^  =  -ix. 

43.  Cylindrical  surfaces.  It  was  seen  in  Art.  42  that  the  locus 
of  a  single  equation  F{x,  y,  z)  =  0  is  a,  surface.  We  shall  now 
discuss  the  types  of  surfaces  which  arise  when  the  form  of  this 
equation  is  restricted  in  certain  ways. 

Theorem.  If  the  equation  of  a  surface  involves  only  tivo  of  the 
coordinates  x,  y,  z,  the  surface  is  a  cylindrical  surface  ivhose  generat- 
ing lines  are  ^mrallel  to  the  axis  ivhose  coordinate  does  not  appear 
in  the  equation. 

Let/(iK,  ?/)  =  0  be  an  equation  containing  the  variables  x  and  y 
but  not  containing  z.  If  we  consider  the  two  equations /(.c,  y)=0, 
z  =  0  simultaneously,  we  have  a  plane  curve  f(x,  y)  =  0  in  the 
plane  z  =  0.  It  (xi,  y-^,  0)  is  a  point  of  this  curve, /(o^i,  ?/i)  =  0. 
The  coordinates  of  any  poiut  on  the  line  x  =  x^,  y  =  yi  are  of  the 
form  a^,  y^,  z.  But  these  coordinates  satisfy  the  equation  f{xy,  y^) 
=  0  independently  of  z,  hence  every  point  of  the  line  lies  on  the 
surface  f(x,  y)  =  0.  It  is  therem^e  generated  by  a  line  moving  par- 
allel to  the  Z-axis  and  always  intersecting  the  curve /(.«,  y)  =  0  in 
the  XF-plane.  The  surface  is  consequently  a  cylindrical  surface. 
In  the  same  w^ay  it  is  shown  that  <f>{x,  z)  =0  is  the  equation  of  a 
cylindrical  surface  whose  generating  elements  are  parallel  to  the 
F-axis,  and  that  F(y,  z)  ~0  is  the  equation  of  a  cylindrical  sur- 
face whose  generating  elements  are  parallel  to  the  X-axis. 

44.  Projecting  cylinders.  A  cylinder  whose  elements  are  per- 
pendicular to  a  given  plane  and  intersect  a  given  curve  is  called 
the  projecting  cylinder  of  the  given  curve  on  the  given  plane. 

The  equation  of  the  projecting  cylinder  of  the  curve  of  inter- 
section of  two  surfaces  F(x,  y,  z)  =  0,  f(x,  y,  z)  =  0  on  the  plane 
2  =  0  is  independent  of  z  (Art.  43).     The  equations  of  this  cylin- 


48  TYPES   OF   SURFACES  [Chap.  IV. 

der  may  be  obtained  by  eliminating  z  between  the  equations  of  the 
curve. 

If  i^  and /are  polynomials  in  z,  the  elimination  may  be  effected 
in  the  following  way,  known  as  Sylvester's  method  of  elimination. 
Since  the  coordinates  of  points  on  the  curve  satisfy  i^=0  and 
/=0,  they  satisfy 

F=  0,  2i^=  0,  z'F=  0,  •",    /=  0,  zf=  0,  ^y  =  0,  ..., 

simultaneously.     If  we  consider  these  equations  as  linear  equa- 
tions in  the  variables  z,  z"^,  z^,  —,  and  eliminate  z  and  its  powers, 
we  obtain  an  equation  R{x',  y)  =  0,  which  is  the  equation  required. 
The  following  example  will  illustrate  the  method. 
Given  the  curve 

2^  +  3  .T2  -f  X  +  2/  =  0,     2  ^2  +  3  2  +  .f  +  ^2  ^  0. 

The  equation  of  its  projecting  cylinder  on  2  =  0  is  found  by  elimi- 
nating 2  between  the  given  equations  and 

z^  -I-  3a-22  +  {x-\-y)z  =  0,     2  z' -\-3  z^  +  (x  4- y^)z  =  0. 

The  result  is 

1  3x    x+y         0 

0      1         3  a;       X  +  y 

2  3      x  +  y^        0 
0      2  3         a;  +  ?/ 

which  simplifies  to 

(y-  —2y  —  xy  =  9  (1  —  2  a;)  {xy'^  -\-x'^  —  x  —  y). 

The  equations  of  the  projecting  cylinders  on  a;  =  0  and  on  y  =  0 
may  be  found  in  a  similar  manner. 

45.    Plane  sections  of  surfaces.     The  equation  of  the  projecting 

cylinder  of  the  section  of  a  surface  F{x,  y,  z)=0  by  a  plane  2  =  A; 
parallel  to  the  XF-plane  may  be  found  by  putting  2  =  A:  in  the 
equation  of  the  surface.  The  section  of  this  cylinder  F(x,y,  k)  =  0 
by  the  plane  2  =  0  is  parallel  to  the  section  by  2  =  k.  Since  paral- 
lel sections  of  a  cylinder,  by  planes  perpendicular  to  the  elements, 
are  congruent,  we  have  the  following  theorem  : 

Theorem.  If  in  the  equation  of  a  surface,  loe  put  z  =  k  and  con- 
sider the  result  as  the  equation  of  a  curve  in  the  plane  2  =  0,  this  carve 
is  congruent  to  the  section  of  the  surface  by  the  plane  z  =  k. 


=  0, 


Art.  46]  CONES  49 

46.  Cones.  A  surface  such  that  the  line  joining  an  arbitrar}^ 
point  on  the  surface  to  a  fixed  point  lies  entirely  on  the  surface  is 
a  cone.     The  fixed  point  is  the  vertex  of  the  cone. 

Theorem.  If  the  equation  of  a  surface  is  homogeneous  in  x,y,  z, 
the  surface  is  a  cone  luith  vertex  at  the  origin. 

'Letf{x,  y,  z)=0  be  the  equation  of  the  surface.  Let  /  be  ho- 
mogeneous of  degree  n  in  {x,  y,  z),  and  let  P^  =(xi,  yi,  Zi)  be  an 
arbitrary  point  on  the  surface,  so  that/(a;i,  ?/i,  Zi)=  0.  The  origin 
lies  on  the  surface,  since  /(O,  0,  0)  =  0.  The  coordinates  of  any 
point  P  on  the  line  joining  P^  to  the  origin  are  (Art.  6) 

x  =  kxi,  yz=ky^,  z^Jcz^,  where  k  = 

r/i,  +  ma 

But  the  coordinates  of  P  satisfy  the  equation,  since 

f{^,  y,  2)  =  /(A;.r„  %i,  kz,)  =  k''f{x^,  y„  z{)=0 

for  every  value  of  k.     Thus,  every  point  of  the  line  OPi  lies  on 
the  surface,  which  is  therefore  a  cone  with  the  vertex  at  the  origin. 

EXERCISES 

1.  Describe  the  loci  represented  by  the  following  equations  : 

(a)  x2  +  y2^4.  ^         iL'  +  ^=l. 

^  ^    4       9 

(b)  y-  =  x.  ..    ^_L^=  1 

^^49' 

(c)  j/  =  sinx.  v' (f)  x(x-l){x-l){x-S)  =  0. 

2.  Describe  as  fully  as  possible  the  locus  of  the  equation  4  x-  +  i/^  =  25  z^. 

3.  Show  that  the  section  of  the  surface  a;'^  +  y-  =  9  ^  by  the  plane  2  =  4 
is  a  circle.     Find  the  coordinates  of  its  center  and  the  length  of  its  radius. 

\      4.   Find  the  equation  of  the  projection  upon  the  plane  2  =  0  of  the  curve 
of  intersection  of  the  surfaces 

2/2+1=0,   {X?  +?/2-  1)2  +  2  2/  =  0. 
*  5.    Show  that  the  section  of  the  surface  x'^z'^  +  a?y'^  =  r-z"^  by  the  plane 
z  =  k  is  an  ellipse.     Find  its  semi-axes.     By  giving  k  a  series  of  values,  de- 
termine the  form  of  the  surface. 

^  6.   Show  that  if  the  equation  of  a  surface  is  homogeneous  in  x  —  ^,  y  —  k, 
z  —  I,  tlie  surface  is  a  cone  with  vertex  at  (/i,  k,  I). 

7.  By  using  homogeneous  coordinates,  show  tliat  the  cylinder /(x,  y,  t)  =0 
can  be  considered  a  cone  with  vertex  at  (0,  0,  1,  0). 


50 


TYPES   OF   SURFACES 


[Chap.  IV. 


47.  Surfaces  of  revolution.  The  surface  generated  by  revolving 
a  plane  curve  about  a  line  in  its  plane  is  called  a  surface  of  revo- 
lution. The  fixed  line  is  called  the  axis  of  revolution.  Every 
point  of  the  revolving  curve  describes  a  circle,  whose  plane  is  per- 
pendicular to  the  axis  of  revolution,  whose  center  is  on  the  axis 
and  whose  radius  is  the  distance  of  the  point  from  the  axis. 

To  determine  the  equation  of  the  surface  generated  by  revolving 
a  given  curve  about  a  given  axis,  take  the  plane  of  the  given  curve 
for  the  X5^-plane  and  the  axis  of  revolution  for  the  X-axis.  Let 
the  equation  of  the  given  curve  in  2  =  0  be  fix,  y)  =  0.  Let 
Pi  =  (a*!,  ?/i,  0),  Fig.  22,  be  any  point  on  the  curve,  so  that  f(xi,  y^)  =  0 


Fig.  22. 


and  let  P  =  (a-,  ?/,  z)  be  any  point  on  the  circle  described  by  Pj. 
Since  the  plane  of  the  circle  is  perpendicular  to  the  X-axis,  the 
equation  of  this  plane  is  .k  =  ^,.  The  coordinates  of  the  center  C 
of  the  circle  are  C  =  (x^,  0,  0);  and  the  radius  CPi  is  y^.  The 
distance  from  C  to  P  is 


,/,  =  V(^i  -  x,y + (y  -  Of +{z-  oy  =  V2/2  +  z^- 

On  substituting  ,  , ; 

a-i  =  X,  ?/i  =  V.y-  -f-  z- 

in  the  equation /(.t'l,  ?/i)=0  we  obtain,  as  the  condition  that  the 
point  P  lies  on  the  surface, 

f{x,  V?^+^)=0, 

which  is  the  desired  ecpmtion. 

In  the  same  way  it  may  be  seen  that  the  equation  of  the  sur- 
face of  revolution  obtained  by  revolving  the  curve /(x,  ?/)=  0  about 

the  F-axis  is  •  /-or   ?     s      n 

/(  Vx^  ■+-  z\  y)  =  0. 


Art.  47]  EXERCISES  51 

EXERCISES 

1.  What  is  the  equation  of  the  surface  generated  by  revolving  the  circle 
x2  +  t/2  =  25  about  the  X-axis  ?  about  the  I'-axis  ? 

y^l.  Obtain  the  equation  of  the  surface  generated  by  revolving  the  line 
2x  +  3y  =  15  about  the  X-axis.  Show  that  the  surface  is  a  cone.  Find  its 
vertex.  What  is  the  equation  of  the  section  made  by  the  plane  x  =  0  ? 
Find  the  equation  of  tlie  cone  generated  by  revolving  the  line  about  the 
I'-axis. 

3.  Why  is  the  resulting  equation  of  the  same  degree  as  that  of  the  gen- 
erating curve  in  Ex.  1,  but  twice  the  degi'ee  of  the  given  curve  in  Ex.  2? 
Formulate  a  general  rule. 

•  4.  What  is  the  equation  of  the  surface  generated  by  revolving  the  line 
y  =  a  about  the  X-axis  ?  about  the  i'-axis  ? 

.  5.  If  the  curve /(.r,  tj)  =  0  crosses  the  x-axis  at  the  point  (xi,  0,  0),  de- 
scribe the  appearance  of  the  surface 

/(./•,  Vy'^  -f-  z^)  —  0  near  the  point  (xi,  0,  0). 

6.  Find  the  equation  of  the  surface  generated  by  revolving  the  following 
curves  about  the  A'-a.-cis  and  about  the  F-axis.    Draw  a  figure  of  each  surface. 

(«)   T  +  -n-  =  l-  ('■)  y-  =  ^^-  (^)    2/  =  sinx. 

4        9 

(6)   ^-f-'=l-  (cO  x2+(^- 1)2  =  4.  if)y=e'. 

a-     0^ 


^ 


/  J    J-  X.  '  . 


CHAPTER   V 

THE   SPHERE 

48.  The  equation  of  the  sphere.  The  equation  of  the  sphere 
having  its  center  at  (a"o,  y^,  Zq)  and  radius  r  is 

(x  -  x,y  +{y  -  y,y  +  {z-  z,f  =  r\  (1) 

or 

ar'  +  2/^  +  2-  -  2  x^x  -2yQy-2zoZ  +  Xq"  +  ?/o'^  +  Zq^  -i~  =  0. 

Any  equation  of  the  form 

a{x''  +  y^  +  z'')  +  2fx  +  2gy  +  2hz  +  k  =  0,     a^O  (2) 

may  be  written  in  the  form 

..{J.(..^J.(..fJ=/l±^^if^^.         (3) 

jf  j2  _j_  ^2  _j_  ^2  _  (^j^.  ^  Q^  |.j^^g  ^g  segi^^  \)y  comparing  with  (1),  to  be 

a  sphere  with  center  at  (  —  — ,   —", )  and  radius 

\     a         a         a) 

V/2  +  ^2  ^  }C-  -  nk 

a 

If  the  expression  under  the  radical  sign  vanishes,  the  center  is 

the  only  real  point  lying  on  the  sphere,  which  in  this  case  has  a 

zero  radius,  and  is  called  a  point  sphere.     If  the  expression  under 

the  radical  is  negative,  no  real  point  lies  on  the  locus,  which  is 

called  an  imaginary  sphere. 

49.  The  absolute.     We  shall  now  prove  the  following  theorem: 

Theorem  I.  All  spheres  intersect  the  plane  at  infinity  in  the 
same  curve. 

In  order  to  determine  the  intersection  of  the  s])liere  and  the 
plane  at  infinity,  we  first  write  the  equation  of  the  sphere  in 
homogeneous  coordinates : 

a  (x-2  +  /  +  z"")  +2fxt  +  2  yyt  +  2  hzt  +  kf^  =  0,     a  ^  0. 

52 


Art.  49]  THE   ABSOLUTE  53 

The  equations  of  the  curve  of  intersection  of  this  sphere  with  the 
plane  at  infinity  are 

t^O,   a;-  +  /  +  2'  =  0.  (4) 

Since  these  equations  are  independent  of  the  coefficients  a,  f,  g, 
h,  k  which  appear  in  the  equation  of  the  sphere,  the  theorem 
follows. 

The  curve  determined  by  equations  (4)  is  called  the  absolute. 
Since  the  homogeneous  coordinates  of  a  point  cannot  all  be  zero 
(Art.  29),  there  are  no  real  points  on  the  absolute. 

The  equation  of  any  surface  of  second  degree  which  contains 
the  absolute  may  be  written  in  the  form 

a  {x^  +  y-  +  z^)  +  (kx  +  hj-\-  VIZ  +  nt)  t  =  0. 

It  a  ^  0,  this  is  the  equation  of  a  sphere  (Art.  48).  If  a  =  0,  the 
locus  of  the  equation  is  two  planes  of  which  at  least  one  is  ^  =  0. 
In  the  latter  case  also,  we  shall  call  the  surface  a  sphere,  since 
its  equation  is  of  the  second  degree  and  it  passes  through  the  abso- 
lute. When  it  is  necessary  to  distinguish  it  from  a  proper  sphere, 
it  will  be  called  a  composite  sphere.  With  this  extended  defini- 
tion, we  have  at  once  the  following  theorem  : 

Theorem  II.  Every  surface  of  the  second  degree  which  contains 
the  absolute  is  a  sphere. 

Any  plane 

ux  4-  vy  4-  wz  -\-  st  —  0, 

other  than  t  =  0,  intersects  the  absolute  in  two  points  whose  coor- 
dinates may  be  found  by  solving  the  equation  of  the  plane  as 
simultaneous  with  the  equations  of  the  absolute.  Any  circle  in 
this  plane  is  the  intersection  of  the  plane  with  a  sphere.  Since 
the  absolute  lies  on  the  sphere,  the  circle  must  pass  through  the 
two  points  in  which  its  plane  intersects  the  absolute.  These  two 
points  are  called  the  circular  points  in  the  plane. 

Evidently  all  the  planes  parallel  to  the  given  one  will  contain 
the  same  circular  points.  The  reason  for  the  designation  circu- 
lar points  is  seen  from  the  fact  that  any  conic  lying  in  any  real 
transversal  plane  and  passing  through  the  circular  points  is  a 
circle,  as  will  now  be  shown.  Since  the  equations  of  the  absolute 
are  not  changed  by  displacement  of  the  axes,  it  is  no  restriction 


54  THE   SPHERE  [Chap.  V. 

to  take  z  =  0  for  the  equation  of  the  transversal  plane.  The 
coordinates  of  the  points  in  which  the  plane  2=0  meets  the 
curve  ^  =  0,  X'  +  y^  +  z^  =  0  are  (1,  i,  0,  0),  (1,  —  i,  0,  0).  A  conic 
in  the  plane  z  =  0  has  an  equation  in  homogeneous  coordinates 
of  the  form 

Ax"-  -f  Bi/^  +  2  Hxy  +  2  Gxt  +  2  Fyt  +  Cf  =  0. 

If  the  points  (1,  /,  0,  0),  (1,  —  i,  0,  0)  lie  on  this  curve, 

A=  B,  11=0. 

But  these  are  exactly  the  conditions  that  the  conic  is  a  circle. 
Conversely,  it  follows  at  once  that  every  circle  in  the  plane  z  =  0 
passes  through  the  two  circular  points  in  that  plane.  A  conic 
in  an  imaginary  plane  will  be  defined  as  a  circle  if  it  passes 
through  the  circular  points  of  the  plane. 

If  the  two  circular  points  in  a  plane  coincide,  the  plane  is  said 
to  be  tangent  to  the  absolute.  Such  a  plane  is  called  an  isotropic 
plane.  The  condition  that  the  plane  ux  +  vt/  +  ivz  -j-  st  =  0  is 
isotropic  is  found,  by  imposing  the  condition  that  its  intersections 
with  the  absolute  coincide,  to  be 

u"  +  V-  +  iv^  =  0.  (5) 

This  equation  is  the  equation  of  the  absolute  in  plane  coordinates. 

EXERCISES 

1.  Write  the  equation  of  a  sphere,  given 

(a)  center  at  (0,  0,  0)  and  radius  r, 
(6)   center  at  (—  1,  4,  2)  and  radius  6, 
(c)    center  at  (2,  1,  5)  and  radius  4. 

2.  Determine  the  center  and  radius  of  each  of  the  following  spheres: 

(a)  x^  +  y'^  +  z'^  +  7x  +  2y  +  z  +  5  =  0. 
lb)  x2  +  2/2  -f  ^2  +  2  X  +  4  ?/  -  6  2  +  14  =  0. 
i    (c)    2(x'^  +  y^  +  z^)-x-2y +  5z  +  6  =  0. 
(d)  x^  +  y^  +  z^+fx^O. 

3.  Find  the  points  of  intersection  of  the  absolute  and  the  plane 

2  x  -  y  +  2  z  +  l^  t  =  0. 

4.  Find  the  coordinates  of  the  points  of  intersection  of  the  line  x  =—  2 
+  I  d,  2/  =  3  -  f  (?,  0  =  -  2  +  i  c?  with  the  sphere  x^  +  y'^  +  z-  +  1=0. 

5.  Show  that  x^  -iry'^  +  z^  =  0  is  the  equation  of  a  cone. 

6.  Find  the  distance  of  the  point  (1,  0,  i)  from  the  origin. 


/ 


Arts.  49-51]       THE   ANGLE   BETWEEN  TWO   SPHERES      55 

7.    Show  that  the  radius  of  the  circle  in  which  ^  =  2  intersects  the  sphere 
-  jJ^  ^  ^'  '^  ^'-^  ^^  imaginary.  iiVtui^     ) 

yv^^B.   Prove  that,  if  (xi,  ij\,  Zi)  is  any  point  exterior  to  the  sphere  (x  —  Xo)^  / ^^.v«-.«-ei^ 
+  {y  —  2/o)-  +  (2  —  2o)-  -  r'^  tlie  expression  (xi  —  Xq)-  +  (iji  -  2/o)^  +  {zi  —  ZoY  \]j^it~^ 
—  r^  is  the  square  of  the  segment  on  a  tangent  from  (xi,  t/i,  Zi)  to  the  point  of  'u  >     ^ 
contact  on  the  sphere. 

/*^  50.    Tangent  Plane.     Let  P  =  (.ri,  ?/,,  Zi)  be  any  point  on  the 

sphere 

a(2;2  +  y2  ^  ^2)  ^  2/a;  +  2  gy  +  2  liz  +  k  =  0. 

The  plane  passing  through  P  perpendicular  to  the  line  joining  P 
to  the  center  of  the  sphere  is  the  tangent  plane  to  the  sphere  at  P. 
\i  is    required    to   find  its   equation.      The   coordinates    of  the 

center  are  (  — •-,  —  •-, )•     The  equations  of  the  line  joining 

\      a        a        a) 

the  center  to  P  are  (Art.  19) 

X  —  a;,  11  —  ?/,  2  —  2, 


-  ^  -  .Ti      -'^-]}x ^x 

a  a  a 


The  ecjuation  of  the  plane  passing 
through  P  and  perpendicular  to  this 
line  is 


Fig.  23. 


If  we  expand  the  first  member  of  this  equation  and  add  to  it 
a{Xi^ +  7/^'^ +  Zi'^)  +  2fXi  +  2  gyi+2hz,+k,  which  is  equal  to  zero 
since  the  point  {x^,  y^,  z^)  lies  on  the  sphere,  we  obtain 

«('^-i^'  +  yiy  +  ^1^)  +  /(■«  +  .i-i)  +  giih  +  y)  +  K^^i  ^z)  +  k  =  o,    (6) 

which  is  the  required  equation  of  the  tangent  plane. 

51.    The  angle  between  two  spheres.     The  angle  between  two 
spheres  at  a  point  P^  on  their  curve  of  intersection  is  defined  as 
equal  to  the  angle  between  the  tangent  planes  to  the  spheres  at  Py 
To  determine  the  magnitude  of  tliis  angle,  let  the  coordinates 
of  Pj  be  (xi,  ?/i,  2i)  and  let  the  equations  of  the  spheres  be 
a(x--  +  v/2  \-  z-')+2fx  -^2  gy  +  2I1Z  +  k  ^i), 
a'(x^  +  2/2  +  Z-)  +  2f'x  -\-2g'y  +  2h'z  +  k'  =  0. 


56  THE   SPHERE  [Chap.  V. 

The  equations  of  the  tangent  planes  to  these  spheres  at  P^  are 

a{x^x  +  y,y  +  z,z)  -\-f{x  +  x^)  +  g{y  +  Ih)  +  h{z  +  ^j)  +  A;  =  0. 
a\x,x  +  y,y  +  z,z)+f\x  +  x,)  +  g\y  +  y,)  +  h'{z  +  ^i)  +  A;'  =  0. 

Since  the  angle  9  between  the  spheres  is  equal  to  the  angle 
between  these  planes,  we  have  (Art.  }5)        ^ 
cog  Q  ^  L  9^%<4>iL  itUiJ-^M^  y^  CA}<^^^ynjtXJ . 

{axi  +f){a'xi  +f)  +  (ayi  +  g)(a'yi  +  (j')  +  (azi+h){a'zi  +  h') 


^(axi+fy  +  {a>ji  +  gy^+iazi  +  hyV{a'xi+f'y^  +  {a'yi+g'y+{a'zi  +  h'y' 
Since  (x■^,  y^,  z^)  lies  on  both  spheres,  this  relation  reduces  to 

2  v/r  +  g^  +  li"  -  akVf'^  +  g""  +  h'''  -  a'k' 

Since  this  expression  is  independent  of  the  coordinates  of  P^,  we 
have  the  following  theorem  : 

Theokem.  Tiro  sjyheres  intersect  at  the  same  angle  at  all  points 
of  their  carve  of  intersection. 

If  ^  =  90  degrees,  the  spheres  are  said  to  be  orthogonal.  The 
condition  that  two  spheres  are  orthogonal  is 

2 //•'  +  2  (/r/'  +  2  hh'  -  aJc'  -  a'k  =  0.  (8) 

52.  Spheres  satisfying  given  conditions.  The  equation  of  a 
sphere  is  homogeneous  in  the  five  coefficients  a,  f,  g,  h,  k.  Hence 
the  sphere  may  he  made  to  satisfy  four  conditions,  as,  for  example, 
to  pass  through  four  given  points,  or  to  intersect,  four  given 
spheres  at  given  angles.  If  the  given  conditions  are  such  that 
a  =  0,  the  sphere  is  composite  (Art.  49). 

EXERCISES 

1.  Prove  that  the  point  (—  3,  1,  —  4)  lies  on  the  sphere  x^  +  y^  -]-  z"^  +  6x 
+  24y-{-Sz  =  0  and  write  the  equation  of  the  tangent  plane  to  the  sphere  at 
that  point. 

2.  Find  the  angle  of  intersection  of  the  spheres  x^  -\-  y-  +  z-  +  x  +  6  y 
+  2  z  +  \)  =  0,  x:^  +  y^  +  z^  +  5  X  +  S  z  +  i  =  0. 

^  3.    Find  the  equation  of  the  sphere  with  its  center  at  (1,  3,  3)  and  making 
an  angle  of  60  degrees  with  the  sphere  x"^  +■  ?/2  +  z-  =  4. 
•  4.    Determine  the  equation  of  the  sphere  which  passes  through  the  points 
(0,0,0),  (0,0,3),  (0,2,0),  (1,2,  1). 


Arts.  52,  53]  LINEAR  SYSTEMS  OF  SPHERES  57 

5.  Determine  the  equation  of  the  sphere  which  passes  through  the  points 
(1,3,2),  (3,2,  -5),  (-1,2,3),  (4,5,2). 

i^'  6.    Write  the  equation  of  the  sphere  passing  through  the  points  (2,  2,  —  1), 
(3,  —  1,  4),  (1,  3,  —2)  and  orthogonal  to  the  sphere 
■X?-  -\-  y-  -V  z'^  —  Z  X  -\-  y  +  z  =  ^. 

*i^.    Write  the  equation  of  the  sphere  inscribed  in  the  tetrahedron  x  =  0, 
y  =  0,  5x+12  2  +  3  =  0,  3x-12y  +  42i=0. 

53.    Linear  systems  of  spheres.     Let 

S'  =  a'  (x^  +  y^  +  z")  +  2fx  +  2  f/'y  +  2  li'z  +  A;'  =  0 

be  the  equation  of  two  spheres.     The  equation 

A,6'  +  X^S'  =  0, 

or         (a.\i  +  a'\,){x'^  +  y^  +  z-^)+2  (f\,  +/A,)  x  +2  (gX,  +  9'X,)y 

+  2  (/iAi  +  /i'A,)  z  +  kX,  +  k'X.  =  0 

also  represents  a  sphere  for  all  values  of  X^  and  A,.  Every 
sphere  of  the  system  Ai*S  +  Ao<S"  =  0  contains  the  curve  of  inter- 
section oi  S  =  0  and  S'  =  0  (Art.  42).  In  particular,  if  aAj  =  —  a'Az, 
the  sphere  A,iS  +  XnS'  =  0  is  composite;  it  consists  of  the  plane  at 
infinity  (which  intersects  all  the  spheres  of  the  system  in  the 
absolute)  and  the  plane 

2 {a'f-  af) x  +  2  {a'g  -  ay') y  +  2 {a'h  -  cM)z  +  a'k  -  ak'  =  0,      (9) 

which  intersects  all  the  spheres  of  the  system  in  a  fixed  circle, 
common  to  S  =  0  and  S'  =  0.  The  plane  (9)  is  called  the  radical 
plane  of  the  given  system  of  spheres. 

It  will  now  be  shown  that  the  radical  plane  is  the  locus  of  the 
centers  of  the  spheres  intersecting  ,S'  =  0  and  *S'  =  0  orthogonally. 
For  this  purpose  let 

ooi^'  +  y'  +  z')  +  2f,x  +  2g,y  +  2h^  +  k,  =  0  (10) 

be  the  equation  of  a  sphere.     It  will  be  orthogonal  to  S  if  (Art 
51) 

2/o/+  2gog  +  2  hji  -  a^k  -  ako  =  0, 
and  to  S'  if 

2/o.r'  +  2 go'/  +  2  /*o^' '  -  ci,k'  -  a%  =  0. 


58  THE   SPHERE  [Chap.  V. 

If  we  eliminate  Jcq  between  these  two  equations,  we  have 
2{a[f-af%  +  2{a'g-ag')go  +  2(a'h-ahyio-{a'k-ak')a,=0,     (11) 

which  is  exactly  the  cendition  that  the  center  (  ,  — ^,  — 

of   the  orthogonal  sphere   lies   in   the   radical   plane  (9).     Con- 
versely, if  ciq,  /o,  go,  Jiq  are  given  numbers  which  satisfy  (9),  a  value 
of  ko  can  be  found  such  that  the  corresponding  sphere  (10)  is 
orthogonal  to  every  sphere  of  the  system  X^S  +  A^^S'  =  0. 
Again,  if 

S"  =  a"  (a;2  +  if  +  z")  +  2/"x  +  2  g"y  +  2  /i"z  +  A;"  =  0 

is  a  sphere  whose  center  does  not  lie  on  the  line  joining  the 
centers  of  8  and  S\  every  sphere  of  the  system 

Ai^  +  A2-S' +  A3>S"  =  0  (12) 

passes  through  the  points  of  intersection  of  the  spheres  »S'=0, 
S'  =  0,  aS"'  =  0. 

Every  sphere   of   the    system    (12)    determined   by  values    of 
Aj,  A2,  A3  for  which 

Aia  +  A,,o'  +A3a"  =  0 

is  composed  of  two  planes  of  which  one  is  the  plane  at  infinity 
and  the  other  passes  through  the  line 

2  {a\f-  af')x  +  2  (a  V/  -  ag')y  +  2  {a'h  -  ah')z  +  a'k  -  ak'  =  0,  (13) 
2  (a"f-af")x  +  2(a"(/  -  ag  ")y  +  2{a"h  -  ali")z + a"k  -  ak"  =  0.  r  a"$'etS' 
This  line  is  called  the  radical  axis  of  the  system  of  spheres  (12), 
By  comparing  equations  (13)  with  (11)  and  the  equation  analo- 
gous to  (11)  for  S"  =  0,  it  may  be  shown  that  the  radical  axis  is 
the  locus  of  centers  of  the  spheres  which  intersect  all  the  spheres 
of  the  system  (12)  orthogonally. 
Now  let 

S'"  =  a'"  (.f2  +  7/2  +  z^)  +  2f"'x  +  2g"'y  +  2  h"'z  +  k'"  =  0 

be  the  equation  of  a  sphere  whose  center  is  not  in  the  plane  de- 
termined by  the  centers  of  /S'  =  0,  /S'  =  0,  ;S"  =  0.  The  condition 
that  a  sphere  of  the  system 

X,S  +  \oS'  +  X^S''  +  XS"'  =  ^ 
is  composite,  is  that  Ai  A2  A3  and  A4  satisfy  the  relation 
Aitt  -\-  Ajtt'  +  Aatt"  -f  A4a"'  =  0. 


Art?  53,  54]       STEREOGRAPHIC  PROJECTION 


59 


The  sphere  orthogonal  to  all  the  spheres  of  the  system  is  in 
this  case  uniquely  determined  by  equations  analogous  to  (W). 
The  center  of  this  orthogonal  sphere  is  called  the  radical  center 
of  the  system.  Through  the  radical  center  passes  one  plane  of 
every  composite  sphere  of  the  system. 

EXERCISES 

•7  1.   Prove  that  the  center  of  any  sphere  of  the  system  \iS  +  X2<S''  =  0  lies 

on  the  line  joining  the  center  of  ;S^  =  0  to  the  center  of  S'  =  0. 

Z-^.    Prove  that  the  line  joining  the  centers  of  the  spheres  ^S*  =  0  and  *S''  =;  0 
r7    is  perpendicular  to  the  radical  plane  of  the  system  \iS  +  X2<S"  =  0. 

'       3.    Show  that  the  radical  axis  of  the  system  \i*S'  +  'hoS'  +  X3<S"'  =  0  is  per- 
■yJ^  pendicular  to  the  plane  of  centers  of  the  spheres  belonging  to  the  system. 

^^  4.    Determine  the  equation  of  the  system  of  spheres  orthogonal  to  the 
.     '  system  Xi^  +  X-^S'  +  f^sS"  =  0. 

1 5.    Show  that  two  point  spheres  are  included  m  the  system  Xi^*  +  MS'  —  0. 
i,     4  6.   Show  that  any  sphere  of  the  system  Xi;S'+X2;S^'  =  0  is  the  locus  of 

a  point,  the  ratio  of  whose  distances  from  the  centers  of  the  two  point 

spheres  of  the  system  is  constant. 

f  7.   If  S  =  0,  S'  =  0,  S"  =  0,  S'"  =  0,  S""  =  0  are  the  equations  of  five 

spheres  which  do  not  belong  to  a  linear  system  of  four  or  less  terms,  show 

that'  the  equation  of  any'  sphere  in  space  can  be  expressed  by  the  equation 

5=SXi5'<'>  =  0.       V£ti/^'y-y^'*^CU<t'ti 

V  54.  Stereographic  projection.  Let  0  be  a  fixed  point  on  the 
surface  of  a  sphere  of  radius  r,  and  let  tt  be  the  plane  tangent  to 
the  sphere  at  the  opposite  end  of  the  diameter  passing  through 
0.  The  intersection  with  tt  of  the  line  joining  0  to  any  point  P^ 
on  the  surface  is  called  the 
stereographic  projection  of 
P,  (Fig.  24). 

To  determine  the  equa- 
tions connecting  the  co- 
ordinates of  Pj  and  its 
projection,  take  the  plane 
TT  for  the  plane  z  =  0,  and 
the  diameter  of  the  sphere 
through  0  for  Z-axis.  The 
equation  of  the  sphere  is 
x'^+y-+z''-2  rz  =  0. 


Fig.  24. 


60  THE   SPHERE  [Chap.  V. 

The  equations  of  the  line  joining  0  =  (0,  0,  2  r)  to  Pi  =  (xi,  y^,  Zi) 
on  the  sphere  are  (Art.  19) 

x_v_z  —  2r 

Xi      ?/,      2i  -  2  r 

To  determine  the  coordinates  (x,  y,  0)  of  P,  the  point  in  which 
OPi  intersects  tt,  we  make  the  equations  of  the  line  simultaneous 
with  2  =  0.     On  solving  for  x,  y,  z  we  obtain 

2  rx,                 2  rvi  A 

x  = i-,    y  = ^L,   z  =  0. 

2r  —  z^  2r  —  Zi 

These  equations  can  be  solved  for  x^,  y^,  z^  by  making  use  of  the 
fact  that,  since  /*i  lies  on  the  sphere, 

^\  +  y\  +  z\-2  rzi  =  0. 
The  results  are 

.   V,  = .    z,  = ^^ ^  '  ' 


.^2-fj/2  +  4r2'    ''      a;2  +  ?/2^4/-2'     '      x'  +  y^  +  Ar''         ^     ^ 

Theorem  I.     The  stereographic  jnojection  of  a  circle  is  a  circle. 

Let  the  equation  of  the  plane  of  the  given  circle  on  the  sphere  be 

Ax  +  By  +  Cz  +  D  =  0. 

The  condition  that  Pi  lies  on  this  circle  is  consequently 

Axi  +  Byi  +  Czi  +  D  =  0. 

If  we  substitute   from  (14)  in  this  equation,  we  obtain  as   the 
equation  of  the  stereographic  projection, 

4  Ar^x  +  4  Br'y  +  2  Or  {x"  +  y'')  +  D  {x"  +  ^^  _^  4  r^)  =  0,      (15) 

which  represents  a  circle  in  the  XF-plane. 

In  particular,  if  the  plane  of  the  given  circle  passes  through  0, 
the  stereographic  projection  of  the  circle  is  composite.  The  con- 
dition that  the  plane 

Ax-[-By-{-Cz-\-D  =  0 

passes  through  0  is 

2rC'+Z)  =  0. 

If  this  condition  is  satisfied,  the  efjuation  of  the  circle  of  projec- 
tion is,  in  homogeneous  coordinates, 

t  {Ax  +By  +  Dt)  =  0. 


Art.  54]  STEREOGRAPHIC   PROJECTION  61 

The  points  of  the  line  t  =  0  correspond  only  to  the  point  0  itself. 
The  line 

is  the  line  of  intersection  of  the  plane  of  the  circle  and  the  plane 
of  projection.     We  have  conseqviently  the  following  theorem  : 

Theorem  II.  The  circles  on  the  sphere  tchich  pass  through  the 
center  of  projection  are  projected  stereographically  into  the  lines  in 
lohich  their  planes  intersect  the  plane  of  p)rojection. 

The  angle  between  two  intersecting  curves  is  defined  as  the 
angle  between  their  tangents  at  the  point  of  intersection.  We 
shall  prove  the  following  theorem  : 

Theorem  III.  Tlie  angle  between  ttvo  intersecting  curves  on  the 
sjjhere  is  eqxial  to  the  angle  between  their  stereog raphic  projections. 

It  will  suffice  if  we  prove  the  theorem  for  great  circles.  For, 
let  C'l  and  C'l  be  any  two  curves  whatever  on  the  sphere  having 
a  point  P'  in  common.  The  great  circles  whose  planes  pass 
through  the  tangents  to  C\  and  Co  at  P'  are  tangent  to  C\  and 
C\,  respectively,  at  P'.  Let  C^,  C^,  and  P,  be  the  stereographic  pro- 
jections of  C'l,  C'2,  and  P.  The  stereographic  projections  of 
the  great  circles  are  tangent  to  C,  and  Co.  respectively,  at  Pi  so 
that  the  angle  between  them  is  the  angle  between  Ci  and 
C2.  If,  then,  the  theorem  holds  for  great  circles,  it  holds  for  all 
intersecting  curves. 

The  condition  that  a  circle  is  a  great  circle  is  that  its  plane 

Ax  +  B;/  +Cz  +  D  =  0 

passes  through  the  center  (0,  0,  r)  so  that 

Cr  +  D  =  0. 

The  equation  (15)  of  the  stereographic  projection  reduces  to 

C  (x2  +  2/2)  +  4  r  (Ax  +  By-  rC)  =  0. 

The  angle  between  two  great  circles  is  equal  to  the  angle  be- 
tween their  planes,  since  the  tangents  to  the  circles  at  their  com- 


62  THE   SPHERE  [Chap.  V. 

mon  points  are  perpendicular  to  the  line  of  intersection  of  their 
planes.     The  angle  6  between  the  planes 

Ax  +  By  +  Cz-Cr  =  0 
and  A'x-hB'y-\-C'z-C'r  =  0 

is  defined  by  the  formula  (Art.  15) 

.  AA'  +  BB'  +  CC  ,.c, 

cosp  =  —  — — —  •  (16) 

The  tangents  to  the  projections 

(7(^2  +  1/)+^  r  {Ax  +  By  -  rC)  =  0, 
C'(x''  +  y')+-i  r{A'x  +  B'y  -  rC)  =  0 

of  the  given  circles,  at  the  point  {x^,  y^)  in  which  they  intersect,  are 

{Cx,  +  2  rA)x  +  (C.vi  +  2  rB)y  +  2  vAx,  4-  2  r%i  -  4  r^C--  0, 
(C'xi  +  2  r^')x  +  (C"?/i  +  2  r5')  ?/  +  2  r.r.^i  +  2  ri3'//i  -  4  r^C  =  0. 

The  angle  <^  between  these  circles  is  given  by  the  formula 

cos  <^  = 

{Cx,  4-  2  rA)  (C'x,  +  2  rA')  +  {Oy,  +  2  rJ3)  ((?>,  +  2  r^Q 


V((7a;i  +  2  ?-^)2+  ((7?/i  +  2  rBf  ^{C'x,  +  2  r^')2+  (C'y,  +  2  ri^')^  • 

By  expanding  this  expression  and  making  use  of  the  fact  that 
(.Tj,  2/i)  lies  on  both  circles,  we  may  simplify  the  preceding  equa- 
tion to 

AA'    +    BB'    +    CC  ,.rr. 

cos<^  =  —  ~ — — =^=1 —  •  (17) 

V^2  +  B'+  C  VA''  +  B"  +  C"2 

From  (16)  and  (17)  we  have  cos  6  =  cos  </>.  We  may  conse- 
quently choose  the  angles  in  such  a  way  that  6  =  <f),  which  proves 
the  proposition. 

The  relation  established  in  Theorem  III  makes  stereographic 
projection  of  great  importance  in  map  drawing. 


s^^ 


CHAPTER   VI 
FORMS  OF  QUADRIC  SURFACES 


55.  Definition  of  a  quadric.  The  locus  of  an  equation  of  the 
seeoud  degree  in  x,  y,  z  is  called  a  quadric  surface.  In  this  chapter 
certain  standard  types  of  the  equation  will  be  considered.  It  will 
be  shown  later  that  the  equation  of  any  non-composite  quadric 
may,  by  a  suitable  transformation  of  coordinates,  be  reduced  to 
one  ot  these  types. 

50.    The  ellipsoid.     The  locus  of  the  equation 

^  +  .^  _l_  ^^  =  1 
a?      b^     c'^ 

is  called  the  ellipsoid.  Since  only  the  second  powers  of  the  varia- 
bles X,  y,  z  appear  in  the  equation,  the  surface  is  symmetrical  as 
to  each  coordinate  plane,  as  to  each  coordinate  axis,  and  as  to  the 
origin. 

The  coordinates  of  the  points  of  intersection  of  the  ellipsoid 
with  the  X-axis  are  found  by  putting  ^  =  ^  =  0  to  be  (±  a,  0,  0). 
Its  intersections  with  the  F-axis  are  (0,  ±  6,  0),  and  with  the  Z-axis 
are  (0,  0,  ±  c).  These  six  points  are  called  the  vertices.  The  seg- 
ments of  the  coordinate  axes  included  between  the  vertices  are 
called  the  axes  of  the  ellipsoid.  The  point  of  intersection  of  the 
axes  is  called  the  center.  The  segments  from  the  center  to  the 
vertices  are  the  semi-axes ;  their  lengths  are  a,  6,  c.  We  shall 
suppose  the  coordinate  axes  are  so  chosen  that  a^h^c  >  0.  The 
segment  joining  the  vertices  on  the  X-axis  is  then  known  as  the 
major  axis ;  that  joining  the  vertices  on  the  Y'-axis  as  the  mean 
axis;  that  joining  the  vertices  on  the  Z-axis  as  the  minor  axis. 

The  section  of  the  ellipsoid  by  the  plane  z  =  A;  is  an  ellipse 
whose  equations  are 


C' 

63 


64 


FORMS  OF  QUADRIC  SURFACES  [Chap.  VI. 


The  semi-axes  of  this  ellipse  are  a*ji  _  ^^  ^-Jl  —  —  .     As  |  A;  |  in- 
creases from  0  to  c,  the  axes  of  the  ellipses  of  section  decrease. 
If  \Jc\=c,  the  ellipse  reduces  to  a  point.     If  j  Zc  j  >  c,  the  ellipse 
of  section  is  imaginary,  since  its  axes  are  imaginary.     The  real 

part  of  the  surface 
j|l^_______^^  therefore    lies    en- 

Yv.,^  tirely  between  the 

planes    z  =  c    and 
z=  —  c. 

In  the  same  man- 
ner, it  is  seen  that 
the  plane  y  =  k' 
intersects  the  sur- 
face in  a  real  ellipse 
if\k'\  <b,  that  the 
ellipse  reduces  to 
a  point  if  I  fc'  I  =  6, 
and  that  it  becomes  imaginary  if  |  A;'  |  >  6.  Finally,  it  is  seen 
that  the  section  x  =  k"  is  a  real  ellipse,  a  point,  or  an  imaginary 
ellipse,  according  as  |  k"  \  is  less  than,  equal  to,  or  greater  than  a. 
The  ellipsoid,  there- 
fore, lies  entirely 
within  the  rectan- 
gular parallelepiped 
formed  by  the  planes 
X  =  a,  ?/=  b,  z  =c; 
x=  -  a,  y=-b, 
z=  —  c,  and  has  one 

point  on  each  of  these  '^^BBIll^^kis^'Ha^® 

planes  (Fig  25). 

If  a  =  b  >  c,  the 

ellipsoid  is  a  surface  of  revolution  (Art.  47)  obtained  by  revolving 

the  ellipse 

x^     y^  _  M 


Fig.  25. 


about  its  minor  axis.     This  surface  is  called  an  oblate  spheroid. 
If  a  >  6  =  c,  the  ellipsoid  is  the  surface  of  revolution  obtained 


Arts.  56,  57]        THE   HYPERBOLOID   OF   ONE   SHEET         65 

by  revolving  the  same  ellipse  about  its  major  axis.  It  is  called  a 
prolate  spheroid. 

If  a  =  b  =  c,  the  surface  is  a  sphere. 

57.    The  hyperboloid  of  one  sheet.     The  surface  represented  by 

the  equation  „        „       „ 

^"  +  ^  _  ?_"  =  1 
a^      6^      c^ 

is  called  an  hjrperboloid  of  one  sheet.  It  is  symmetric  as  to  each  of 
the  coordinate  planes,  as  to  each  of  the  coordinate  axes,  and  as  to 
the  origin. 

The  section  of  the  surface  by  the  plane  z  =  k  is   an   ellipse 
whose  equations  are 

This  ellipse  is  real  for  every  real  value  of  k.     The  semi-axes  are 


«v 


which  are  the  smallest  when  A-  =  0,  and  increase  without  limit  as 
I  k  I  increases.  For  no  value  of  k  does  the  ellipse  reduce  to  a 
point. 

The  plane  y  =  k'  intersects  the  surface  in  the  hyperbola 

=  1,     y  =  k: 


If  \k'  \  <  b,  the  transverse  axis  of  the  hyperbola  is  the  line 
2  =  0,  y  =  k',  and  the  conjugate  axis  is  x  =  0,y  =  k']  the  lengths 

of  the  semi-axes  are  a\  1 ,  c\  1 .    As  I  A;'  I  increases  from 

zero  to  b,  the  semi-axes  decrease  to  zero.      When  j  k'  ]  =  b,  the 

equation  cannot  be  put  in  the  above  form,  but  becomes  ~ =  0 

a  2      c^ 

and  the  hyperbola  is  composite  ;  it  consists  of  the  two  lines 

"^  +  '  =  0,  y=b;   "^-"  =  0,  y  =  b; 
a     c  a     c 


66  FORMS   OF  QUADRIC   SURFACES  [Chap.  VI. 

when  k'  =  —  b,  the  hyperbola  consists  of  the  lines 

0,  y  =  -b. 


-  +  -  =  0,  y  =  -b;  ---■■ 

a      c  a      c 


These  four  lines  lie  entirely  on  the  surface.  If  \k'  \  >  b,  the 
transverse  axis  of  the  section  is  x  —  0,  y  =  k'  and  the  conjugate 
axis  is  z  =  0,  y  =  k'.     The  lengths  of  the  serai-axes  are 


62  '  \  Jf. 

They  increase  without  limit  as  k'  increases. 

The  plane  x  =  k"  intersects  the  surface  in  the  hyperbola 
y^  z^ 


bH\ 


cn\- 


A:"2 


1,  x=k'\ 


Of-  J  \        a" 

If  I  A;"  I  <  a,  the  transverse  axis  of  this  hyperbola  is  2  =  0,  .-c  =  A:". 
The  section  on  the  plane  x  =  a  consists  of  the  two  lines  _-^ 

"7'-0'-3 


b 


0,     a;  =  a ;   "^ =  0,     a;  =  a.  J     -'rz: 


c  be 

The  section  on  the  plane  x  =  —  a  consists  of  the  lines 

|  +  ?  =  0,     .r  =  -a;  f-?  =  0,     x  =  -a. 
be  be 

If  I  k"  I  >  a,  the  line  y  =  0,  x  =  k"  is  the  transverse  axis  and  2=0, 
x  =  k"  is  the  conjugate  axis. 
As  I  k"  I  increases,  the  lengths 
of  the  semi-axes  increase  with- 
out limit.  The  form  of  the 
surface  is  indicated  in  Fi<r.  26. 


Fig.  26. 


Arts.  57,  58]       THE   HYPERBOLOID   OF   TWO   SHEETS        67 


If  a  =  b,  the  hyperboloid  is  the  surface  of  revolution  obtained 
by  revolving  the  hyperbola 


---  =  1,    y  =  o 


about  its  conjugate  axis. 

58.    The  hyperboloid  of  two  sheets.     The  locus  of  the  equation 


d?'     h"     c^ 


=  1 


is  called  an  hyperboloid  of  two  sheets.     It  is  symmetric  as  to  each 
of  the  coordinate  planes,  the  coordinate  axes,  and  the  origin. 


Fig.  27. 
The  plane  z  =  k  intersects  the  surface  in  the  hyperbola 

^2  y2 


a^fl  +  l' 


W{1  + 


A;2 


=  1,     z  =  k. 


The   transverse  axis  is  y  =  0,   z  =  k,  for   all  values  of   k.     The 

They   are 


I       ^         /        k"^ 
lengths   of   the    semi-axes    are  a\/l  H — ,  &\/l  +  — 


smallest  for  A:  =  0,  namely  a  and  b,  and  increase  without  limit  as 
I  k  I    increases.     The   hyperbola   is   not   composite    for   any   real 
value  of  k. 
The  plane  y  =  k'  intersects  the  surface  in  the  hyperbola 

=  1,     y  =  k'. 


aHl  + 


i'-'S) 


68 


FORMS  OP  QUADRIC  SURFACES 


[Chap.  VI. 


The  transverse  axis  is  z  =  0,  y  =  k'.  The  conjugate  axis  is 
X  =  0,  y  =  k'.  If  k'  =  0,  the  lengths  of  the  semi-axes  are  a  and  c ; 
they  increase  without  limit  as  k'  increases. 

The  plane  x  =  k"  intersects  the  surface  in  the  ellipse 

+  -7777^ r  =  l,     ^  =  k". 


r 


This  ellipse  is   imaginary  if    |  k"  |  <  a.     If    ]  Zc"  |  =  a,  the   semi- 
axes  are  zeio;  they  increase  without  limit  as  k"  increases. 


If  b  =  c,  the  hyperboloid  of  two  sheets  is  the  surface  of  revolu- 
tion obtained  by  revolving  the  hyperbola 


-  ^  =  1,     z  =  0 

62 


about  its  transverse  axis. 


59.   The  imaginary  ellipsoid.  The  surface  defined  by  the  equa- 
tion 

x^     y^  z^  _  _^ 

a^      b^  c^ 

is  called  an  imaginary  ellipsoid.  Since  the  sum  of  the  squares  of 
three  real  numbers  cannot  be  negative,  there  are  no  real  points  on 
it. 


Arts.  59,  60]         THE   ELLIPTIC  PARABOLOID  69 

EXERCISES 

^  1.  By  translating  the  axes  of  coordinates,  show  that  the  surface  defined 
by  the  equation  2  x-  +  3  j/'^  +  4  2^  _  4  a;  _  e  y  -j-  IH  ^  +  16  =  0  is  an  ellipsoid. 
Find  the  coordinates  of  the  center  and  the  lengths  of  the  semi-axes. 

2.  Classify  and  describe  the  surface  x-  +  j/2  _  4  a;  —  3  ?/  +  10  2  =  20  —  z^. 

3.  Show  that  the  surface  2  x^  —  3  z-  —  5  ^  =  7  —  2  j/'^  is  a  surface  of  revo- 
lution.    Find  the  equations  of  the  generating  curve. 

^^4.  On  the  hyperboloid  of  one  sheet  x^  4-  y-  —  2^  =  1,  find  the  equations 
of  the  two  lines  which  pass  through  the  point  (1,  0,  0)  ;  through  (—  1,  0,  0). 

h   5.    Classify  and  plot  the  loci  defined  by  the  following  equations  : 
(a)  9  x2  +  16  2/2  -f  25  z^  =  1,  (d)  y;i  +  ^2  _  4  ^2  _  25, 

(6)  4  x2  -  9  2/2  -  16  22  =  25,  (e)   x2  -f  4  2/2  -f-  ^2  _  9^ 

(c)   4  x2  -  16  2/2  -^  9  5;2  =  25,  (/)    x2  -f-  4  2/2  -f-  9  ^2  +  8  =  0, 

60.    The  elliptic  paraboloid.     Tlie  Icx'us  of  the  equation 

is  called  an  elliptic  paraboloid.  The  surface  is  symmetric  as  to 
the  planes  x  =  0  and  ?/  =  0  but  not  as  to  z  =  0.  It  passes  through 
the  origin,  and  lies  on  the  positive  side  of  2  =  0  if  n  is  positive 
and  on  the  negative  side  if  n  is  negative.  In  the  following  dis- 
cussion it  will  be  assumed  that  n  is  positive.  If  n  is  negative,  it 
is  necessary  only  to  reflect  the  surface  on  the  plane  2:  =  0. 

The  section  of  the  paraboloid  by  the  plane  2;  =  A;  is  an  ellipse 
whose  semi-axes  are  aVlJ  nk  and  6 V2  >(A-,  respectively.  If  A;  <  0, 
the  ellipse  is  imaginary.  If  A-  =  0,  the  ellipse  reduces  to  a  point, 
the  origin.  As  A'  increases,  the  semi-axes  of  the  ellipse  increase 
without  limit. 

The  section  of  the  paraboloid  by  the  plane  y  =  A'  is  the 
parabola 

For  all  values  of  k'  these  parabolas  are  congruent.  As  k'  in- 
creases, the  vertices  recede  from  the  plane  y=  0  along  the  parabola 

y-=2nz,     x  =  0. 

62 


V 


70 


FORMS  OF  QUADRIC  SURFACES 


[Chap.  VI. 


Fig.  28. 

li  a  —  b,  the  paraboloid  is 
tlie  surface  of  revolution 
generated   by    revolving  the 

parabola    '—  =  2  nz,     y  =  0 
about  the  Z-axis. 

61.  The  hyperbolic  parab- 
oloid. The  surface  defined 
by  the  equation 


The  sections  by  the  planes  x  = 
k"  are  the  congruent  parabolas 


r_9 


=  'Z  vz 


7.M2 

'^,     x  =  k". 


Their   vertices   describe   the  pa- 
rabola 


—  =  2  nz, 


y  =  o. 


The  form  of  the  surface  is  in- 
dicated by  Fig.  28. 


is  called  an  hyperbolic  paraboloid.  The  surface  is  symmetric  as  to 
the  planes  x  =0  and  ?/  =  0,  but  not  as  to  2  =  0. 

As  before,  let  it  be  assumed  that  n  >  0.     The  plane  z  =  k  inter- 
sects  the  surface  in  the  hyperbola 

^^     -      -^^^      =1    z  =  7c 
a^2nk     b^2nk       ' 

If  A;  >  0,  the  line  x  —  0,  z  =  k  \s  the  transverse  axis  and  y  =  0, 
z  =  k  is  the  conjugate  axis.  If  k  <  0,  the  axes  are  interchanged. 
The  lengths  of  the  semi-axes  increase  without  limit  as  |A;|  increases. 


"^, 


(I-  J  -J. 


Arts.  61,  62] 


THE   QUADRIC   CONES 


71 


When  k  =  0,  the  section  of  the  paraboloid  consists  of  the  two 
lines 

^  +  ^  =  0,2  =  0;    --^=0,    z  =  0. 
ah  a      b 


Fig.  29. 


The  sections  of  the  surface  by  the  planes  y  =  k'  are  the  con- 
gruent parabolas 


V^lf'i 


a-  0- 


'.'.n 


0 


The  vertices  of  these 
parabolas  describe  the 
parabola 

^  =  -  2  H2r,  a-  =  0. 

The  sections  by  the 
planes  x  =  A'"  are 
congruent  parabolas 
whose  vertices  de- 
scribe the  parabola 


^jLtf^i* 


62.   The  quadric  cones.     The  cone  (Art.  46) 


^    1. 


=  0 


72  FORMS  OF  QUADRIC   SURFACES  [Chap.  VI. 

is  called  the  real  quadric  cone.  Its  vertex  is  at  the  origin.  The 
section  of  the  cone  by  the  plane  2;  =  c  is  the  ellipse 

The  cone  is  therefore  the  locus  of  a  line  which  passes  through  the 
origin  and  intersects  this  ellipse. 

If  a  =  b,  the  surface  is  the  right  circular  cone  generated  by  re- 
volving the  line  -  =    ,  y  =  0  about  the  Z^axis. 
a      c 

The  equation 

o?      b"^      & 

represents  an  imaginary  quadric  cone.  There  are  no  real  points 
on  it  except  the  origin. 

63.  The   quadric    cylinders.      The   cylinders    (Art.  43)   whose 
equations  are 

a^      b"^         '   «2      b"^         ^    a?      b"^ 

are  called  elliptic,  hyperbolic,  imaginary,  and  parabolic  cylinders, 

respectively,  since  the  sections  of  them  by  the  planes  z  =  k  are 
congruent  ellipses,  hyperbolas,  imaginary  ellipses,  and  parabolas, 
respectively. 

64.  Summary.     The   surfaces  discussed   will  be   enumerated 

again  for  reference. 


Ellipsoid. 

(Art.  56) 

x'y-      z^      -, 

Hyperboloid  of  one  sheet. 

(Art.  57) 

3.2               y2               Z-    _^ 

'a?     1?      ?~    ' 

Hyperboloid  of  two  sheets. 

(Art.  58) 

x^      tf-  I  2;^  _ 

1.      Imaginary  ellipsoid. 

(Art.  59) 

?,  +  -^  =  2'«- 

Elliptic  paraboloid. 

(Art.  60) 

Art.  64]  SUMMARY  73 

^  =  2  nz.     Hyperbolic  paraboloid.  (Art.  61) 

a?     b^ 

^^t-t=,0.     Keal  quadric  cone.  (Art.  62) 

a^      b"^     c^ 

—  +  —  -\ —  =  0.     Imaginary  quadric  cone.  (Art.  62) 
a-     b^      c^ 

—  ±  ^  =  ±  1 ;   y^  =  2px.     Quadric  cylinders.  (Art.  63) 
a*     &* 

EXERCISES 

Classify  the  following  surfaces : 

1.  4  22-6x2  + 2  2/2  =  3. 

2.  x-^  +  3  2/2  +  5  a;  +  2  2/  +  7  =  0. 

3.  x2  +  3  2/2  +  4  X  -  2  2  =  0. 

4.  4  x2  +  4  2/2  -  3  ^2  =  0. 

5.  2  22  _  x2  -  3  2/2  -  2  X  -  12  2/  =  15. 

6.  x2  -  2  j/2  -  6  2/  -  6  2  =  0. 

7.  Find  the  equation  of  and  classify  the  locus  of  a  point  which  moves  so 
that  (a)  the  sum  of  its  distances,  {b)  the  difference  of  its  distances  from  two 
fixed  points  is  constant.     Take  the  points  (±  a,  0,  0). 

8.  Find  and  classify  the  equation  of  the  locus  of  a  point  which  moves  so 
that  its  distance  from  (a,  0,  0)  bears  a  constant  ratio  to  its  distance  (a)  from 
the  plane  x  =  0  ;  (6)  from  the  Z-axis. 

9.  Show  that  the  locus  of  a  point  whose  distance  from  a  fixed  plane  is  al- 
ways equal  to  its  distance  from  a  fixed  line  perpendicular  to  the  plane  is  a 
quadric  cone. 

10.  A  line  moves  in  such  a  way  that  three  points  fixed  on  it  remain  in 
three  fixed  planes  at  right  angles  to  each  other.  Show  that  any  other  point 
fixed  on  the  line  describes  an  ellipsoid.  (Sug.  Find  the  direction  cosines  of 
the  line  in  terms  of  the  coordinates  of  the  point  chosen,  and  substitute  in 
formula  (1),  Art.  3.) 


CHAPTER  VII 
CLASSIFICATION  OF  QUADRIC  SURFACES 

65.    Intersection  of  a  quadric  and  a  line.     The  most  general  form 
of  the  equation  of  a  quadric  surface  is  (Art.  55) 
F{x,  y,  z)  =  ax^  +  hy'^  +  cz-  +  2fyz  +  2  gzx  +  2  hxy 

+  2  te  +  2  ?/i?/  +  2  //z  4-  fZ  =  0.  (1) 

We  shall  suppose,  unless  the  contrary  is  stated,  that  the  coeffi- 
cients are  all  real,  and  that  the  coefficients  of  the  second-degree 
terms  are  not  all  zero. 

To  determine  the  points  of  intersection  of  a  given  line  (Art.  20) 
ic  =  a-o  +  Ar,     y  =  Po  +  f^r,     z  =  z^  +  vr  (2) 

with  the  quadric  (1),  substitute  tlie  values  of  x,  y,  z  from  (2)  in 
F{x,  y,  z)  and  arrange  in  powers  of  r.     The  result  is 

Qf-  +  2Rr  +  S  =  0,  (3) 

in  which 

Q  =  a\2  +  hfx^  +  cv^  +  2/^v  +  2  ^vX  +  2  /^A/.,  (4) 

R  =  {axo + /i//o  4-  gzo  +  0  A  +  (/i.i'o  +  f>yo  +.fzo  -\-m)ix  +  (gxo  +/2/0 + cZq +n)v 
IfdF,    ,    dF      ,   dF 

=  -    A  +  /A  H 1 

2  \dxQ         dyo         Ozq 
S  =  F{Xo,  yo,  Zo). 

The  roots  in  r  of  equation  (3)  are  the  distances  from  the  point 
Pq  =  (^o>  2/o>  ^o)  01^  the  line  (2)  to  the  points  in  which  this  line 
intersects  the  quadric. 

If  Q  :^0,  equation  (3)  is  a  quadratic  in  r.  If  Q  =  0,  but  R  and 
S  are  not  both  zero,  (3)  is  still  to  be  considered  a  quadratic,  with 
one  or  more  infinite  roots.  If  Q  =  72  =  /S'  =  0,  (2)  is  satisfied  for 
all  values  of  r  and  the  corresponding  line  (2)  lies  entirely  on  the 
quadric.     We  have,  consequently,  the  following  theorems  : 

Theorem  I.  Every  line  ickkh  does  not  lie  on  a  given  quadric 
surface  has  two  {distinct  of  coincident)  points  in  common  loith  the 
surjace.  ,' 

74 


Arts.  65,  66] 


DIAMETRAL  PLANES,   CENTER 


75 


Theorem  IL  If  a  given  line  has  more  than  two  points  in  common 
with  a  given  qnadric,  it  lies  entirely  on  the  quadric. 

For,  if  (3)  is  satisfied  by  more  than  two  values  of  r,  it  is  satis- 
fied for  all  values. 

66.  Diametral  planes,  center.  Let  P,  and  P2  be  the  points  of 
intersection  of  the  line  (2)  with  the  quadric.  The  segment  P1P2 
is  called  a  chord  of  the  quadric. 

Theorem  I.  T7ie  locus  of  the  middle  point  of  a  system  ofp)arallel 
chords  of  a  quadric  is  a  plane. 

Let  r,  and  r2  be  the  roots  of  (3)  so  that  P^Pi  =  r^  and  PqPo  =  r^. 
The  condition  that  Pq  ^s  the  middle  point  of  the  chord  P1P2  is 

PoP,  +  P,P,  =  0, 
or 

ri  +  r2  =  0.  ^' 

Hence,  from  (4),  we  have  JVAt+A^    so 

'{axo  +  hy^  +  gz^  +  0'^  +  (^'-^'o  -|-  ^J'h  +.ho  +  m)fi 

+  (.7-^0  +  fUo  +  <^^o  +  n)  V  =  0'.  '^ "  '  (5) 

If,  now,  \,  fx,  V  are  constants,  but  x^,  yg,  Zq  are  allowed  to  vary, 

the  line  (2)  describes  a  system  of  parallel  lines.     The  locus  of  the 

middle  points  of  the  chords  on  these  lines  is  given  by  (5).     Since 

(5)  is  linear  in  Xq,  y^,  Zq,  this  locus  is  a  plane. 

Such  a  plane  is  called  a  diametral  plane.        ^'    /■  ."^  ^  '   r''  ^  X' 

Theorem  II.  All  the  diametral  planes  of  a  quadric  have  at  least 
one  (finite  or  infinite)  jwint  in  common. 

For  all  values  of  A,  ix,  v  the  plane  (5)  passes  through  the  inter- 
section of  the  planes 

ax  +  hy  -\-  gz  +  I  =  0," 

hx+by+fz-\-m  =  0,-  (6) 

gx  +  fy  -f  cz  -j-  n  =  0. 

In  discussing  the  locus  determined  by  (6),  it  will  be  convenient 
to  put,  for  brevity, 


D  = 


a    h    g 

a    h    I 

a  g  I 

h    g   I 

h    b  f 

,  N  = 

h    b    m 

,  M= 

h  f  m 

,  L  = 

b    f    m 

9  f   c 

9   .f    n 

g  c  n 

fen 

(7) 


^G 


6  QUADRIC  SURFACES  [Chap.  VII. 

If   D  =^  0,   the   planes    (6)    intersect   in  a  single    finite   point 

(Art.  26) 

L  M  N  ,^. 

If  this  point  (xq,  y^,  Zq)  does  not  lie  on  the  surface,  it  is  called 
the  center  of  the  quadric.  It  is  the  middle  point  of  every  chord 
through  it.  If  the  point  (xq,  y^,  Zq)  does  lie  on  the  surface,  it  is 
called  a  vertex  of  the  quadric.  In  either  case  the  system  of  planes 
(5)  is  a  bundle  with  vertex  at 


f _L  M   _N\ 
\     D' D'       D)' 


If  Z)  =  0,  but  L,  M,  N  are  not  all  zero,  the  planes  (6)  intersect 
in  a  single  infinitely  distant  point,  the  homogeneous  coordinates 
of  which  are  found,  by  making  (6)  homogeneous  and  solving,  to 
be  (L,  —  M,  N,  0).  The  system  of  planes  (5)  is  a  parallel  bundle. 
The  quadric  is,  in  this  case,  said  to  be  non-central. 

If  the  system  of  planes  (6)  is  of  rank  two  (Art.  35),  the  planes 
determine  a  line;  the  diametral  planes  (5)  constitute  a  pencil  of 
planes  through  the  line.  If  this  line  is  finite  and  does  not  lie 
on  the  quadric,  it  is  called  a  line  of  centers ;  if  it  is  finite  and  does 
lie  on  the  quadric,  it  is  called  a  line  of  vertices.  If  the  system  is 
of  rank  one,  the  diametrical  planes  coincide.  If  each  point  of  this 
plane  does  not  lie  on  the  quadric,  it  is  called  a  plane  of  centers ;  if 
every  point  of  the  plane  lies  on  the  quadric,  it  is  called  a  plane  of 
vertices. 

Example.     Find  the  center  of  the  quadric 

The  equations  (6)  for  determining  the  center  are 
a:  +  2?/  +  2;+l  =  0,       x  +  2//  +  2  +  l  =  0,         x-|-2y  —  z  —  1=0, 

from   which  x  +  2i/  =  0,  2+1=0.     This   line   is  a  line  of  centers  unless 
d  =  —  1,  in  which  case  it  is  a  line  of  vertices. 

EXERCISES 

2 
1.    Find   the  coordinates  of   the   points   in   which  the  line  x=l  +  -r, 

2  T 

y  =  —  2 r,  2=— 1+-  intersects  the  quadric  x^  +  Sy"^  —  4:z'^  +  'iz~2  y— 

o  o 

6  ::=0. 


Arts.  66,  67]       QUADRIC   REFERRED  TO   ITS  CENTER       77 

Find  which  of  the  following  quadrics  have  centers.  Locate  the  center 
when  it  exists. 

2.  z^-2y'2  +6z^  +  12xz  -U=  0. 

>^  3.  2z^+y^  —  z^  —  2xz  +  ixy  +  4:yz  +  2y-4z~i  =  0. 

4.  xy  +  yz  +  zx  —  X  +  2  y  —  z  —  9  =  0. 

5.  2  x^  +  5  y'^  +  z"  -  4  xy  -  2  X  -  i  y  -  8  =  0. 
f^6.  x^  —  xz  —  yz  —  z  =  0. 

#7.   x^  +  y^  +  z'  —  2yz  +  2xz  —  2xy  —  x  +  y  -  z  =0. 
P  e.   x"^  +  4  y-  +  z^  -  i  yz  —  2  xz  +  i  xy  +  \0  x  +  ^  y  -  7  z  +  15  =  0. 
■)Ikk      9.   Show  that  any  plane  which  passes  through  the  center  of  a  quadric  is 
a  diametral  plane. 

10.  Let  Pi  and  P^  be  two  points  on  an  ellipsoid,  and  let  0  be  its  center. 
Prove  that  if  Pi  is  on  the  diametral  plane  of  the  system  of  chords  parallel  to 
OP2,  then  P2  is  on  the  diametral  plane  of  the  system  of  chords  parallel  to 
OPi. 

67.  Equation  of  a  quadric  referred  to  its  center.  If  a  quadric 
has  a  center  {xq,  y^,  Zq),  its  equation,  referred  to  its  center  as  origin, 
may  be  obtained  in  the  following  way  : 

If  we  translate  the  origin  to  the  center  by  putting 
x  =  x'  +  Xo,         y  =  y'-\-yo,         z  =  z'  +  z^, 
the  equation  F{x,  y,z)=  0  is  transformed  into 
ax"  +  by''  +  cz'^  +  2fy'z'  +  2  gz'x'  +  2  hx'y'  +  2{ax,  +  hy,  + 
gZo+  l)x'  +  2(/tXo  +  %o  +fio  +  wi)^'  +  2(gXo+fyo  +  cz^  +  n)z'  +  S  =  0 
wherein,  as  in  Eq.  (4),  S  =  F(xq,  yo,  z^). 

Since  (xq,  y^,  Zq)  is  the  center,  it  follows  from  (6)  that 

axo -\- hyo  +  gZo-\- I    =0, 

hxo  +  byo  +JZo  +  7n  =  0,  (8) 

gxo  +  fyo  +  c'Zf,  i- n  =0, 

so  that  the  coefficients  of    x',  y',  z'  are  zero,  and  the  equation  has 
the  form  (after  dropping  the  accents) 

ax'  +  by'  +  cz'  +  2fyz  +  2  gzx+2hxy  +  S  =  0.-^  (9) 

The  function  S  =  F{xq,  yo,  z^)  may  be  written  in  the  form 

S  =  F(xo,  yo,  Zo)  = 

Xo{axo  +  hyo  +  gzo  +  I)  +  yo{hxo  +  byo  -\-fzo  +  m)  +  Zo(gxo  4-./>/o+  c^o  + 

n)  +  Ixo  +  myo  +  nzo  +  d. 


78 


QUADRIC  SURFACES 


[Chap.  VII. 


Hence,  from  (8)  we  have 

^S  =  Ixo  +  m?/o  +  nzo  +  d.  (10) 

By  eliminating  Xo,  y^,  2;,,  from  (8)  and  (10)  we  obtain  the  relation 


h 
b 

f 
m 


I 

m 
n 
d-S 


=  0. 


This  equation  may  be  written  in  the  form 


a     h     g 

h     b    f 

S  = 

9    f    c 

h 

b 

f 
m 


9 

/ 

f 

m 

c 

n 

?i 

d 

Denote   the   right-hand   member  of    this   equation   by   A 
coefficient  of  S  is  D  (Eq.  7).     Hence 

DS  =  A, 
or,  if  D=^0, 


\:\\ 


The 


V.^-     ,./v*^ 


D 


(11) 


If  2)^0  and  A=0,  it  follows  from  (9)  and  (11)  that  the 
quadric  is  a  cone  (Art.  46).  The  vertex  of  the  quadric  is  the 
vertex  of  the  cone.  v'l. 

If  A  =  0  and  S  ^0,  then  Z>  =  0.  Since  {Xq,  iJq,  Zq)  was 
assumed  to  be  a  finite  point,  it  follows  that  L  =  M=  N=  0  so 
that  the  surface  has  a  line  or  plane  of  centers. 

If  A  =  0  and  S  =  D  =  0,  then  from  (9)  the  surface  is  com- 
posite.   Every  point  common  to  the  component  planes  is  a  vertex. 

The  determinant  A  is  called  the  discriminant  of  the  given 
quadric.  If  A  =  0,  the  (piadric  is  said  to  be  singular.  If  A  ^  0, 
the  quadric  is  non-singular. 

68.  Principal  planes.  A  diametral  plane  which  is  perpendic- 
ular to  the  chords  it  bisects  is  called  a  principal  plane. 

Theorem.     If  the  coefficients  in  the  equation  of  a  quadric  are 
real,  and  if  the  qnadric  does  not  have  the  plane  at  in-finity  as  a  com-  ^ 
ponent,  the  quadric  has  at  least  one  real,  finite,  principal  plane. 


Arts.  68,  69]        THE   DISCRIMINATING   CUBIC  79 

The  condition  that  the  diametral  plane  (5) 

(ciA  +  hix  +  gv)x  +  {h\  +  bfj.-\-  fv)y  +  (fi'A  +fiJ.-\-  cv)z  + 

IX  +  ?/iyu,  +  wv  =  0 

is  perpendicular  to  the  chords  it  bisects  is  (Art.  14) 

aX  +  hfi  +  gv  _  hX  +  ^i^  +>  _  fifA  +//a  +  cv  ,^2\ 

A  /A  V 

If  we  denote  the  common  value  of  these  fractions  by  k,  equa- 
tions (12)  may  be  replaced  by 

(a  —  A-)A  +  h/x  -\-  gv  =  0, 
hx+{b-k)ti+fv=0,  (13) 

gX+ffji-{-(c-k)v  =  0. 

The  condition  that  these  equations  in  A,  fx,  v  have  a  solution  other 

■  ■-'J-',  O   vv^-'  '-- 

=  0,  (14) 


than  0,0,  0  is  -^^--o  vv-  •  - 


a  —  k        h  g 

h        h-k        f 
9  f       c  -  A; 

or,  developed  and  arranged  in  powers  of  k, 

Jc^  -(a  +  6  +  c)k^+{ab  +  bc-^ca-f^-  g"-  -  h'')k  -D  =  0,    (15) 

where  B  has  the  same  meaning  as  in  (7).     This  equation  is  called 
the  discriminating  cubic  of  the  quadric  F{x,  y,  z)  =  0. 

To  each  real  root,  different  from  zero,  of  the  discriminating 
cubic  corresponds,  on  account  of  (13),  (12),  and  (5),  a  real  finite 
principal  plane.  Our  theorem  will  consequently  be  proved  if  we 
show  that  equation  (15)  has  at  least  one  real  root  different  from 
zero.     The  proof  will  be  given  in  the  next  article.  . 

69.  Reality  of  the  roots  of  the  discriminating  cubic.  We  shall 
first  prove  the  following  theorem  : 

Theorem  I.      TJie  roots  of  the  discriminating  cubic  are  all  real. 

Let  fcj  be  any  root  of  (15)  and  let  Aq,  /xq,  vq  (not  all  zero)  be  values 
of  A,  fx,  V  that  satisfy  (13)  when  k  =  k^.  If  A;  is  a  complex  number, 
Aq,  /hq,  Vq  may  be  complex.     Let 

^0=^1  +  iX\,  ixq  =  ixi  +  ifx\,  vo  =  vi  +  iv\, 
where  ?'  =  V  —  1  and  Aj,  A'l,  lx■^,  fx\,  vj,  v\  are  real. 


80  QUADRIC  SURFACES  [Chap.  VII. 

Substitute  fcj  and  these  values  of  X.^,  /xq,  vq  for  A;,  A,  fi,  v  in  (13), 
multiply  the  resulting  equations  by  Aj  —  i\\,  fj.^  —  iix\,  vj  —  iv\,  re- 
spectively, and  add.     The  result  is 

(A^2  +  A  V  +  /.i^  +  tx.\'  +  V,'  +  v'r)  k,  =  (Ai^  +  A'l^)  a  +  (/xi^  +  ^Lt'i^)  6 
+  (vi^  +  v'l^)  c  +  2  (/.iv:  +  /xVv'i)/  +  2  (viA,  +  v'A'O  gr 
+  2(Ai/xi+AVi)/i- 
The  coefficient  of  A:i  is  real  and  different  from  zero.     The  number 
in  the  other  member  of  the  equation  is  real.     Hence  k^  is  real. 
Since  A;i  is  any  root  of  (15),  the  theorem  follows. 

Theorem  II.  Not  all  the  roots  of  the  discriminating  cubic  are 
equal  to  zero. 

The  condition  that  all  the  roots  of  (15)  are  zero  is 

a-\-h  +  c  =  0,ah  +  hc  +  ca-f-  —  g'^-h^=Q,D  =  0. 

Square  the  first  member  of  the  first  equation,  and  subtract  twice 
the  first  member  of  the  second  from  it.     The  result  is 
a?  +  62  _,_  ^2  ^  2/2  +  2  ^2  _^  2  7^2  =  0. 

Since  these  numbers  are  real,  it  follows  that 

a  =  6  =  c  =  /  =  gr  =  /i  =  0 ; 

but  if  these  conditions  are  satisfied,  the  equation  of  the  quadric 
contains  no  term  in  the  second  degree  in  x,  y,  z,  which  is  contrary 
to  hypothesis  (Art.  65). 

70.  Simplification  of  the  equation  of  a  quadric.  Let  the  axes  be 
transformed  in  such  a  way  that  a  real,  finite  principal  plane  of  the 
quadric  F{x,  y,  z)  =0  is  taken  as  x  =  0.  Since  the  surface  is  now 
symmetric  with  respect  to  a;  =  0  (Art.  68),  the  coefficients  of  the 
terms  of  first  degree  in  x  must  all  be  zero.  Hence  the  equation 
has  the  form 

ax-2  +  6(/2  +  cz"^  +  2fyz  +  2my +  27iz +  d  =  0. 

Moreover,  fiTtO,  since  otherwise  x  =  0  would  not  be  a  principal 

plane  (Art.  68). 

Now  let  the  planes  y  =  0,  2  =  0  be  rotated  about  the   X-axis 

2  f 
through  the  angle  6  defined  by  tan  2  0  =  — •—-  •     This  rotation  re- 

b  —  c 


.    ^RTS.  70,  71]        CLASSIFICATION   OF  QUADRICS  81 

duces  the  coefficient  of  yz  to  zero,  and  the  equation  has  the  form 
"\  i.  iL'    a'x'-[-h'y-  +  c'z'  +  2m'y-\-2n'z  +  d'  =  0,  (16) 

■wherein  a'  ^  0,  but  any  of  the  other  coefficients  may  be  equal  to  zero. 

71.    Classification  of  quadric  surfaces.     Since  the  equation  of  a 
quadric  can  always  be  reduced  to  the  form  (16),  a  complete  classi- 
fication can  be  made  by  considering  the  possible  values  of  the  co- 
V    efficients. 

--  I.    Let  both  h'  and  c'  be  different  from  zero.     By  translation  of 

the  axes  in  such  a  way  that  f  0,  — — ,    — —  j  is  the  new  origin, 

the  equation  reduces  to 

If  d"  ^  0,  divide  by  d"  and  put 

a  h  c' 

the  signs  being  so  chosen  that  a,  b,  c  are  real.     This  gives  the  fol- 
lowing four  types : 

t  +  t  +  t  =  i.     Ellipsoid.  (Art.  56) 

or      Ir      c- 

/p2  y2         2^ 

— \-- =  1.     Hyperboloid  one  sheet.         (Art.  57) 

a?      \r      <? 

—  —  -^  — ^  =  1-     Hyperboloid  two  sheets.       (Art.  58) 
a^      Ir      c- 

-  —  ^  —  ^=1.     Imaginary  ellipsoid.  (Art.  59) 

€?      h-     & 

If  d"  =  0,  the  reduced  forms  are 

'—  +  77  +  —  =  0.     Imaginary  cone.  (Art.  62) 

w-      IP-      c 

t  +  t     t  =  0.     Real  cone.  (Art.  62) 

or      ¥      c^  ^ 

IL   Letc'  =  0,  6'^0. 

If  n'  ^t  0,  by  a  translation  of  axes,  the  equations  may  be  re- 
duced to 

a'x^  +  b'y^  +  2  n'z  =  0. 


82 


QUADRIC  SURFACES 


[Chap.  VII. 


This  equation  takes  the  form 


■^  —  2  n2. 


Elliptic  paraboloid. 
Hyperbolic  paraboloid. 


or  

according  as  a'  and  6'  have  the  same  or  opposite  signs. 
If  n'  =  0,  the  equation  may  be  reduced  to 

a'.-r^  +  6y  +  rf"  =  p. 
If  (/"  ^  0,  this  may  be  written  in  the  form 

^.2 


(Art.  60) 
(Art.  61) 


±^±1 


0.     Quadric  cylinder. 


(Art.  63) 


and  if  d"  =  0, 


Pair  of  intersecting  planes. 


III.    Let6'  =  c'  =  0.     Equation  (16)  is  in  this  case 
aV  +  2m'?/  +  2n'^  +  d'=0. 

If  m  and  n  are  not  both  zero,  since  the  plane  2  m'y  +  2  ?i'2;  +  d' 
=  0  is  at  right  angles  to  x  =  0,  we  may  rotate  and  translate  the 
axes  so  that  this  plane  is  the  new  ?/  =  0.  The  equation  of  the  sur- 
face becomes 

;r2  =  2  my.     Parabolic  cylinder.  (Art.  63) 

If  m'  and  n'  are  both  zero,  we  have, 
if  d'^0,     x^ ^k'^  —  0.     Two  parallel  planes. 

if  (V  =  0,  x"^  =  0.     One  plane  counted  twice. 

72.  Invariants  under  motion.  A  function  of  the  coefficients  of 
the  equation  of  a  surface,  the  value  of  which  is  unchanged  when 
the  axes  are  rotated  and  translated  (Arts.  36  and  37),  is  called  an 
invariant  under  motion  of  the  given  surface.  It  will  be  shown 
that  the  expressions 

7=a  +  6  +  c, 

J^  6c  +  ca  +  oh  —p 


D  = 


a     h 

9 

h     6 

f 

9    f 

c 

A  = 


-9'- 

h\ 

a     h 

9 

I 

h     6 

f 

m 

9    f 

c 

n 

I     m 

n 

d 

Arts.  72,  73]  INVARIANCE  OP  /,  J,  AND  D  83 

formed  from  the  coefficients  of  the  equation  (1)  of  a  quadric  are 
invariants  under  motion. 

73.  Proof  that  /,  /,  and  D  are  invariants.  When  the  axes  ar^ 
translated  (Art.  36),  the  coefficients  of  the  terms  in  the  second 
degree  in  the  equation  of  a  quadric  are  unchanged.  Hence  /,  ,/, 
and  D  are  unchanged. 

Since  the  equations  of  rotation  (Art.  37)  are  linear  and  homo- 
geneous in  X,  y,  z,  x\  y',  z',  the  degree  of  any  term  is  not  changed 
by  these  transformations,  so  that  a  term  of  the  first  degree  can- 
not be  made  to  be  of  the  second,  nor  conversely.  Suppose  the 
expression 

f{x,  y,  z)  =  ax^  +  2  hxy  -f  &/  +  2  (/x^  +  2fyz  +  cz"^ 

is  transformed  by  a  rotation  into 

f\x',  y',  z')  =  a'x'^  +  2  h'x'y'  +  h'y'""  +  2  g'x'z'  +  2f'y'z'  +  c'z'\ 

Now  consider  the  function 

<^(.^-,  y,  z)  =  f(x,  y,  z)  -  k{x'  +  y'  +  z'). 

The  expression  x^  +  y^  +  z^  is  the  square  of  the  distance  of  a  point 
(x,  y,  z)  from  the  origin,  and  will  therefore  remain  of  the  same 
form  .r'2  +  y'^-f-z'^  by  the  transformation  of  rotation  (Art.  37). 

If,  then,/(x,  y,  z)  is  changed  into  f'(x',  y',  z'),  cf){x,  y,  z)  will  be 
changed  into 

<f>'{x\  y'  z')  =f'{x',  y\  z')  -  A:(x'==  +  y"  +  z"). 

If  k  has  such  a  value  that  </>  is  the  product  of  two  linear  factors  in 
X,  y,  z,  then,  for  the  same  value  of  k,  the  expression  <j>'  will  be  the 
product  of  two  linear  factors  in  x',  y',  z'.  The  condition  that  <^  is 
the  product  of  two  factors  is  that  its  discriminant  vanishes,  that 
is 

a  —  k         h  g 

h         h-k        /     =0, 

g  f        c-k 

which,  developed  in  powers  of  k,  is  exactly  the  equation  of  the  dis- 
criminating cubic  (Art.  68) 


84  QUADRIC   SURFACES  [Chap.  VIL 

Similarly,  the  condition  that  <f>'  is  the  product  of  two  linear  fac- 
tors is  Ji^-I'k^-\-J'k-D'  =  0, 

where  /',  J',  and  D'  are  the  expressions  I,  J,  and  D  formed  from 
the  coefficients  oif'(x',  y',  z'). 

These  two  equations  have  the  same  roots,  hence  the  coefficients 
of  like  powers  of  k  must  be  proportional.  But  the  coefficient  of  k? 
is  unity  in  each,  hence, 

/'  =  7,  J'  =  J,  D'  =  D, 

that  is,  I,  J,  D  are  invariants. 

From  the  theorem  just  proved  the  following  is  readily  obtained : 
"'   Theorem.      When  the  axes  are  transformed  in  such  a  loay  that 
the  coefficients  of  xy,  yz,  and  zx  are  all  zero,  the  coefficients  of  x^,  y^^ 
and  z^  are  the  roots  of  the  discriminating  cubic. 

For,  if  the  equation  of  the  quadric  has  been  reduced  to 
a'x^  +  by  +  c'z2  +  2rx  +  2  m'y  +  2  u'z  +  d'  =  0, 
the  discriminating  cubic  is 

A;3  _  (a'  +  6'  +  c')k  +  {a'b'  +  b'c'  +  c'a.')k  -  a'b'c'  =  0. 

The  roots  of  this  equation  are  a',  b',  and  c'.  This  proves  the 
proposition. 

From  the  theorem  just  proved,  the  following  criteria  immedi- 
ately follow : 

If  two  roots  of  the  discriminating  cubic  are  equal  and  different 
from  zero,  the  quadric  is  a  surface  of  revolution,  and  conversely. 

If  all  three  roots  of  the  discriminating  cubic  are  equal  and 
different  from  zero,  the  quadric  is  a  sphere. 

If  A  T^  0,  and  a  root  of  the  discriminating  cubic  is  zero,  the 
quadric  is  non-central. 

If  two  roots  of  the  discriminating  cubic  are  equal  to  zero,  the 
terms  of  second  degree  in  the  equation  of  the  quadric  form  a 
perfect  square. 

74.    Proof  that  A  is  invariant.     It  will  first  be  proved  that  A  is 
invariant  under  rotation.     The  reasoning  is  similar  to  that  in 
Art.  73.     Let 
F(x,  y,  z)=  ax""  +  by""  +  cz^  +  2fyz  +  2  gzx  +  2hxy  +  2lx  +  2  my 

+  2nz  +  d  =  0 


Art.  74]  PROOP^  THAT   A   IS  INVARIANT  85 

be  the  equation  of  the  given  quadric.     Let  this  equation  be  trans- 
formed by  a  rotation  into 

F{x\  y',  z')  =  a'x'^  +  b'y'^  +  cY'  +  2f'y'z'  +  2  g'z'x'  +  2  h'x'y'  +  2 1'x' 

+  2  m'y'  +  2  n'z'  +  d'  =  0. 

This  rotation  transforms  the  expression 

$  {x,  y,  z)  =  F(x,  y,  z)  -  k  (x'  -{-y^  +  z-'  +  l) 
into  $'(ic',  y',  z')  =  F'{x',  y',  z')  -  k  {x'^  +  y"'  +  z""  +  1). 

The  discriminants  of  $  and  $'  are,  respectively, 


a  —  k 

h 

9 

h 

b-k 

f 

9 

f 

c  —  k 

I 

m 

n 

I 

m 

n 

d-k 


and 


a'-^■  h'  g'  V 

h'  b'-k  f  m' 

g'  f  c'-k  n' 

V  m'  n'  d'-k 


The  roots  of  the  quartic  equations  in  A;  obtained  by  equating  these 
discriminants  to  zero  are  equal ;  since  a  value  of  k  which  makes 
<I>  =  0  singular  also  makes  4>'  =  0  singular  and  conversely  (Art.  67). 
Hence,  since  the  coefficient  of  k*  in  each  equation  is  unity,  the 
constant  terms  are  equal ;  that  is,  A  =  A'.  Hence,  A  is  invariant 
under  rotation. 

In  order  to  prove  that  A  is  invariant  under  translation,  let  the 
axes  be  translated  to  parallel  axes  through  (x^,  y^,  z^).  The  equa- 
tion of  the  quadric  becomes  (cf.  Art.  67) 

F{x',  y',  z')  =  ax'^  -f  by'^  +  cz'^  +  2fx'y'  +  2  gy'z'  +  2  hz'x' 

+  2  (ax,  +  hy,  +  gz,  +  /)  x'  +  2  {hx,  +  by,  +fz,  +  w)  y' 
+  2  (gx,  +  /2/0  +  cz,  +  n)z'  +  S  =  0, 

where  S  =  F{x,,  y„  z^.     The  discriminant  of  F'(x',  y',  z')  is 

a  h  g  axo+hyo+gZ(,  +  l 

h  b  f  hxo  +  byo  +  fzo  +  m 

9  f  c  gxo+  fyo  +  czo+n 

axo+hyo+gzo+l  hxo+byo+fzo+m  gxo+fyo+czo+n  S 

Multiply  the  first  column  by  x„  the  second  by  y,,  the  third  by  z,, 
and  subtract  their  sum  from  the  last  column.  In  the  resulting 
determinant,  multiply  the  first  row  by  x„  the  second  by  y,,  the 
third  by  z„  and  subtract  their  sum  from  the  last  row.  Finally 
divide  the  first  row  and  column  each  by  Xq,  the  second  row  and 
column  each  by  y,,  and  the  third  row  and  column  each  by  z,. 


86  QUADRIC  SURFACES  [Chap.  VII. 

The  resulting  determinant  is  A.  Hence  A'  =  A,  so  that  A' is  inva- 
riant under  translation.  Since  A  is  invariant  unaer  both  transla- 
tion and  rotation,  it  is  invariant  under  motion. 

75.  Discussion  of  numerical  equations.  In  order  to  determine 
the  form  and  position  of  a  quadric  with  a  given,  numerical  equa- 
tion, it  is  advisable  to  determine  the  standard  form  (Art.  71)  to 
which  the  equation  of  the  given  quadric  may  be  reduced,  and  the 
position  in  space  of  the  coordinate  axes  for  which  the  equation 
has  this  standard  form.  For  this  purpose  the  roots  k^,  ko,  ks  of 
the  discriminating  cubic  and  the  value  of  the  discriminant  A 
should  first  be  computed. 

A.  If  all  the  roots  ki,  k^,  k^  are  different  from  zero,  the  three 
principal  planes  may  be  determined  as  in  Art.  68.  If  these  planes 
are  taken  as  coordinate  planes,  the  equation  reduces  to  (Art.  67, 
Eq.  11 ;  Art.  73) 

^•i^"  +  hy'  +  hz'  +  TT^  =  0. 

B.  If  one  root  k-^  is  zero,  two  finite  principal  planes  may  be 
determined  as  before.  Let  these  be  taken  as  cc  =  0  and  y  =  0. 
At  least  one  intei*section  of  the  new  Z-axis  with  the  surface  is  at 
infinity.  If  this  axis  does  not  lie  on  the  surface,  and  does  meet 
the  surface  in  one  finite  point,  the  axes. should  be  translated  to 
this  point  as  origin.  The  equation  of  the  surface  now  has  the 
form 

k\x''-\-k.^/^-^2n"z  =  0. 
Since 


A  = 


it  follows  that 


^\ 

0 

0 

0 

0 

k. 

0 

0 

0 

0 

0 

n 

0 

0 

n" 

0 

k\k2 


If  the  new  Z-axis  lies  on  the  quadric,  or  if  it  has  no  finite  point 
in  common  with  it,  any  point  on  the  new  .Z-axis  may  be  chosen  for 
origin  and  the  equation  takes  the  form 

k^x"  +  k.^f  +  S=0, 


Art.  75]        DISCUSSION  OF  NUMERICAL  EQUATIONS         87 

where  (Art.  67) 

S  =  lXo+  viyo  +  nzo  +  d, 

and  (xq,  ijo,  Zq)  are  the  old  coordinates  of  the  new  origin. 

C.  If  two  roots  of  the  discriminating  cubic  are  zero,  the  terms 
of  the  second  degree  in  the  original  equation  form  a,  perfect  square, 
so  that  the  equation  of  the  surface,  referred  to  the  original  axes, 
is  of  the  form  ^^■'':: 

(ax  +  /3y  +  yzf  +  2  Ix  +  2  m?/  -f  2  nz  +  d  =  0, 

or    (ax  +  /3y+yz  +  Sf  +  2(1  -  a8)x  +  2(m  -  /3B)y  +  2(n  -  y8)z 

e^,  +d-8'  =  0.  (17) 

If  the  planes  ax  +  /3y  -\-yz  +  8  =  0, 

2(1  -  aS)x  +  2(m  -  ^8)y  +  2(n  -  y8)z  +  d  -  8^  =  0 

are  not  parallel,  we  may  choose  8  so  that  they  are  perpendicular. 
The  first  term  of  (17)  is  proportional  to  the  square  of  the  distance 
of  the  point  (x,  y,  z)  from  the  plane 

ax+  (iy  +  yz+8  =  0. 

.  .  The  remaining  terms  of  (17)  are  proportional  to  the  distance  to 
the  second  plane.  If  these  planes,  with  the  appropriate  value  of 
8,  are  chosen  as  x  =  0,  y  =  0,  the  equation  reduces  to 


(«2  +  ^2  +  y--yf  +  2V(?  -  a8f  +  (m  -  138)'+ (n  -  y8y  x  =  0. 

If  the  two  planes  are  parallel,  8  may  be  so  chosen  that 
l-a8  =  0,     m  -  ^8  =  0,     n  -  y8  =  0. 
The  equation  now  becomes 

(u-^+fS^  +  y^f  +  d    -8^=0, 

wherein  ax  -\-  /3y  -\-  yz  -}-  8  —  0  is  the  new  y  =  0. 

Example  1.     Discuss  the  equation 

a;2  _  2  j/2  ^_  6  2>  +  12  xz  -  16  X  -  4  y  -  36  z  +  &2  =  0. 

Tlie   equations  determining   tlie  center  are  x  +  6  z  —  S  =  0,  2y  +  2  =  0, 
6  X  -\-  6  z  —  18  =  0,  from  which  the  coordinates  of  the  center  are  (2,—  1,  1). 
The  invariants  are  /  =  5,  J"  =  —  44,  Z>  =  GO,    A  =  1800. 
Hence,  the  discriminating  cubic  is 

A;3  _  5  ^.2  _  44  ;t  -  60  =  0. 
Its  roots  are  ki  =  10,  k^  ——  2,  kz  =  —  3.    The  transformed  equation  is 
10  a;2  _  2  2/2  _  3  ^2  +  30  =  0. 


88  QUADRIC  SURFACES  [Chap.  VII. 

The  direction  cosines  of  the  new  axes  through  (2,  —  1,  1)  are  found,  as  in 
Art.  68,  by  giving  k  the  values  10,  —  2,  —  3,  to  be 

O  Q  S  2 

——,  0,    -^;  0,   1,  0;    — ^,  0,    — =• 
V13  Vl3  Vl3  Vl3 

The  surface  is  an  hyperboloid  of  one  sheet. 

Example  2.     Discuss  the  quadric 

11  x2  +  10  2/2  +  6  22  -  8  J/.S  +  4  2X  -  12  xy  +  72  X  -  72  t/  +  36  2  +  150  =  0. 
The  discriminating  cubic  is 

k^  -  27  k^  +  180  yfc  -  324  =  0. 

Its  roots  are  3,  6,  18.  A  =  ^  3888.     The  surface  is  an  ellipsoid. 

The  equations  for  finding  the  center  are 

11  X  -  6  y  +  2  z  +  36  =  0,     -  6  x  +  10  y  -  4  2;  -  36  =  0, 

2  X  -  4  ?/  +  6  .2  +  18  =  0. 

The  coordinates  of  the  center  are  (—  2,  2,-1).  The  direction  cosines  of 
the  axes  are 

1        2        2.         21        _2.         _22        1 

3'     3'     5  '         3'      3'  3   '  3'     3'  ^* 

The  equation  of  the  ellipsoid  referred  to  its  axes  is 
3x2  _^  (^yi  +  is^i-  12. 

Example  3.     Discuss  the  quadric 
Sx^  -y2  +  2z^  +  6yz  -4:ZX--2  ry  -  14  x  +  4  y  +  20  2  +  21  =  0. 

The  discrinainating  cubic  is 

^-3_4i.-2_  13^.^  19  =  0. 

Its  roots  are  approximately  1.2,  5.7,  —  2.9.     A  =  0.     The  surface  is  a  cone. 
The  equations  for  finding  the  vertex  are 

Sx-y-2z-7=0,  -  X  -  //  +  3  2  +  2  =  0,  -  2  x  +  3  x  +  2  z  +  10  =  0. 
The  coordinates  of  the  vertex  are  (1,  —  2,  —  1).  The  direction  cosines  of 
the  axes  are  approximately 

.8.  .4,  .5;   .6,  -4,    -.7;  0,  .6,   -  .4. 
The  equation  of  the  cone  referred  to  its  axes  is  approximately 
1.2  x2  +  5.7  2/2  _  2.9  22  =  0. 

Example  4.     Discuss  the  quadric 

4  x^  +  y"^  +  z^  -  2  yz  +  i  xz  —  i  xy  —  8  X  +  i  z  +  1  =  0. 

This  equation  may  be  written  in  the  form 

(2x-2/  +  z  +  5)2=  (8  +  4  5)x-2  52/-(4-25)2-7  +52.       . 

If  5  =  —  1,  the  planes  2Xf-y  +z-l  -0  and  4x  +  2y  — 6z  — 6--0  are 
perpendicular.  If  we  take  these  planes  as  ?/'  =  0  and  x'  =  0,  the  equation  of 
the  surface  reduces  to  6  2/2  =  V56  x.     The  surface  is  a  parabolic  cylinder. 


Art.  75]     DISCUSSION  OF  NUMERICAL  EQUATIONS     89 

EXERCISES 

Discuss  the  quadrics: 

1.  3  a;2  +  2  2/2  +  22  _  4  xy  -  4  yz  +  2  =  0. 

2.  x^-y^  +  2z^-2yz  +  4iXZ  +  Axy-2x-4:y-l=0. 
''§.  6  y2  +  8  2^  +  6  yg  +  6  xz  +  2  xy  +  2  X  +  4  y  -  2  z  -  1  =  0. 

4.  4x2  +  y2-8z2  +  8yz-4xz  +  4xy-8x-4y  +  4z  +  4  =  0. 

6.  3x2  +  2y2  +  2z2-4y3-2zx  +  2xy-6x  +  2y  +  2z-12  =  0. 
'6.  6x2-2z2-6yz-6x3-2xy  +  2x  +  4y  +  2z  =  0. 

7.  4x2  +  4y2  +  22-4yz-4xz  +  8xy-6y  +  6z-3  =  0, 
/^  8.  x''  -  yz  +  xz  -  xy  +  X  +  y  +  2  z  -  2  =  0. 

■■>'   9.  3  y2  +  6  yz  +  6  xy  -  2  X  +  2  z  +  4  =  0. 

10.  3  x"  +  3  y2  +  z2  +  2  yz  +  2  xz  -  2  xy  -  7  X  +  y  +  6  z  -  7  =  0. 

11.  3  x2  -  5  y2  +  15  z2  -  22  yz  +  14  xz  -  14  xy  +  2  X  -  10  y  +  6  z  -  5  =  0. 

12.  x2  -  y2  -  2  z2  -  4  yz  +  2  xy  -  2  y  +  2  z  =  0. 

13.  x"  -  6  yz  +  3  zx  +  2  xy  +  X  -  13  z  =  0. 

14.  x2  -  2  y2  +  z2  -  4  zx  -  12  xy  +  4  y  +  4  z  -  9  =  0. 
16.  x2  +  2  y2  +  2  z2  +  2  xy  -  2  X  -  4  y  -  4  z  =  0. 

16.  3  x2  +  y2  +  z2  +  yz  -  3  zx  -  2  xy  +  2  X  +  4  y  +  2  z  =  0. 

17.  For  what  values  of  c  is  the  surface 

5  x2  +  3  y2  +  cz2  +  2  xz  +  15  =  0 
a  surface  of  revolution? 

18.  Determine  d  in  such  a  way  that 

x2  +  y'^  +  5z2  +  2ya  +  4x2-4xy  +  2x  +  2y  +  d  =  0 
is  a  coue. 


CHAPTER   VIII 

SOME   PROPERTIES   OF   QUADRIC  SURFACES 

76.  Tangent  lines  and- planes.  If  the  two  points  of  intersection 
of  a  line  and  a  quadric  coincide  at  a  point  Pq,  the  line  is  called  a 
tangent  line  and  Pq  the  point  of  tangency.  If  the  surface  is  sin- 
gular, it  is  supposed  in  this  definition  that  Pq  is  not  a  vertex. 

Theorem.  TJie  locus  of  the  lines  tangent  to  the  quadric  at  Pq  is 
a  plane. 

Let  the  equation  of  the  quadric  be 

F{x,  y,  z)  =  ax^  +  by'^  +  cz^  +  ^fy^  +  2  gzx  +  2  hxy 

+  2lx  +  2  my  +  2  7iz  -j-  d  —  0,  (1) 

and  let  the  equation  of  any  line  through  P „  =  (-^'o  ^o  ^o)  ^^ 
(Art.  20) 

X  =  Xo  +  Xr,     y  =  y,-}-  fxr,     z  =  Zq  +  vi:  (2) 

Since  Pq  lies  on  the  quadric,  F(xq,  y^,  Zq)  =  0.  Hence,  one  root  of 
equation  (3),  Chapter  VII,  which  determines  the  intersections  of 
the  line  (2)  with  the  quadric  (1),  is  zero.  The  condition  that  a 
second  root  is  zero  is  P  =  0,  or 

A  {axQ  +  hy^  -f-  (/^^  +  /)  +  /a  (JiXa  +  hy^  +  Jzq  +  m) 

+  v{gxQ  +  /Vo  -+-  ''^0  +  »0=  0-       (3) 
If  we  substitute  in  (.3)  the  values  of  A,  p.,  v  from  (2),  we  obtain 

(x  —  Xo)(axo  +  hyo  +  gz^  +  l)  +  (y  -  yo)(hXo  +  byo  +  J'Zq  +  m) 

+  {z-  Zo)  (gxa  +  fyo  +  cz^  +  7i)  =  0,        (4) 

which  must  be  satisfied  by  the  coordinates  of  every  point  of  every 
line  tangent  to  the  quadric  at  Pq.  Conversely,  if  (x,  y,  z)  is  any 
point  distinct  from  Pq,  whose  coordinates  satisfy  (4),  the  line  de- 
termined by  (ic,  y,  z)  and  Pq  is  tangent  to  the  surface  at  Pq. 
Since  (4)  is  of  the  first  degree  in  (.t,  y,  z),  it  is  the  equation  of  a 
plane.     This  plane  is  called  the  tangent  plane  at  Pq. 

90 


Arts.  76,  77]  TANGENT   PLANE   NORMAL  FORM  91 

The  equation  (4)  of  the  tangent  plane  may  be  simplified.  Mul- 
tiply out,  transpose  the  constant  terms  to  the  second  member,  and 
add  Ixq  +  m>jQ  -f  tiZq  +  d  to  each  member  of  the  equation.  The 
second  member  is  F(Xa,  r/^,  Zq),  which  is  equal  to  zero,  since  Pq  lies 
on  the  quadric.  The  equation  of  the  tangent  plane  thus  reduces 
to  the  form 

axxo  +  byi/o  +  czZq  +  /{yz^  +  zy^)  +  f/  {zx^  +  -rzc)  +■  h  (-^V/o  +  V^o) 

+  l{x  +  .i-o)  +  m  (y  +  .vo)  +  7^  (z  +  z,)-{-d  =  0.  (5) 

*  This  equation  is  easily  remembered.  It  may  be  obtained  from 
the  equation  of  the  quadric  by  replacing  x^,  y'^,  z^  by  xx^^  yy^,  zZq  ; 
2  yz,  2  zx,  2  xy  by  yz^  +  zyo,  zx^  +  xz^,  xy^  +  yx^  ■  and  2x,2y,2zhy 
^  +  3*0)  y  +  ?/o)  2!  4-  ^0,  respectively. 

77.    Normal  forms  of  the  equation  of  the  tangent  plane.     The  equa- 
>'^ion  of  the  tangent  plane  to  the  central  quadric 

ax''  +  by""  +  cz^  =  1  (6) 

at  the  point  (x^,  y^,  Zq)  on  it  is 

axxQ  +  byyo  +  czZq  =  1, 

Let  the  normal  form  of  the  equation  of  this  plane  (Art.  13)  be 

A.T  -\-  fj.y  +  vz  =  p,  (7) 


so  that 

A                   u        ,           V 

-  =  aXo,     ^  =  bye,     -  =  czq. 

P                P                P 

Since  (xg, 

Ih, 

Zq)  lies  on  the  quadric,  we  have 

from  which 

«.V  +  ^Jlfo^  +  t'^o'  =  1, 

X2  „2  „2 

^4-^^-  +  ^=^^  (8) 

a       b        c 

Conversely,  if  this  equation  is  satisfied,  the  plane  (7)  is  tangent 
to  the  quadric  (6). 

By  substituting  the  value  of  p  from  (8)  in  (7),  we  have 


Xx  +  f,y  +  rz  =  J^  +  i  +  -, 
^  a        b        c 

which  is  called  the  normal  form  of  the  equation  of  the  tangent 
plane  to  the  central  quadric  (6). 


92         PROPERTIES  OF  QUADRIC  SURFACES       [Chap.  VIII. 

It  follows  from  (8)  that  the  necessary  and  sufficient  condition 

that  the  plane 

ux  -{-  vy  +  wz  =  1 

is  tangent  to  the  quadric  (6)  is  that 

t  +  t  +  }!^==l.  VlV'  (9) 

a        b        c 

This  equation  is  called  the  equation  of   the  quadric  (6)  in  plane 

coordinates. 

Again,  if 

ax^  +  bif=2  nz  (10) 

is  the  equation  of  a  paraboloid  (Arts.  60  and  61),  it  is  proved  in  a 
similar  way  that  the  normal  form  of  the  equation  of  the  tangent 
plane  to  the  paraboloid  is 

X.  +  ,y  +  v.  =  -i(f  +  f)  (11) 

and  that  the  condition  that  the  plane 

iix  -{-  vy  -\-wz  =  1 
is  tangent  to  the  paraboloid  is 

'iV  -  +  —  =  0.  (12) 

a        b        n 

Equation  (12)  is  the  equation  of  the  paraboloid  in  plane  coordinates. 

78.  Normal  to  a  quadric.  The  line  through  a  point  P(,  on  a 
quadric,  perpendicular  to  the  tangent  plane  at  P^,  is  called  the 
normal  to  the  surface  at  Pq. 

It  follows  from  equation  (4)  that  the  equations  of  the  normal 
at  Po  to  the  quadric  F{x,  y,  z)  =  0  are 

X  —  Xn  _  y  —  lk  _  2:  —  Zq 


aXf^  +  hyf^  +  gZf^  +  I     hxo  +  byo  +  fz^  +  m     gxo  +  fyo  +  cz^  +  n 


(13) 


EXERCISES 


1.  Show  that  the  point  (1,  -  2,  1)  Hes  on  the  quadric  x^  —  y^  +  z^  + 
4  yz  +  2  zx  +  xy  —  X  +  y  +  z  +  12  =  0.  Write  the  equations  of  the  tangent 
plane  and  the  normal  line  at  this  point. 

2.  Show  that  the  equation  of  the  tangent  plane  to  a  sphere,  as  derived  in 
Art.  76,  agrees  with  the  equation  obtained  in  Art.  50. 


Arts.  78. 79]         RECTILINEAR  GENERATORS  93 

3.  Prove  that  the  normals  to  a  central  quadric  ar?  -\-  by^  +  cz-  =  1,  at  all 
points  on  it,  in  a  plane  parallel  to  a  principal  plane,  meet  two  fixed  lines, 
one  in  each  of  the  other  two  principal  planes. 

4.  Prove  that,  if  all  the  normals  to  the  central  quadric  ax^  +  hif  -}-  cz^  =  l 
intersect  the  X-axis,  the  quadric  is  a  surface  of  revolution  about  the  JT-axis. 

5.  Prove  that  the  tangent  plane  at  any  point  of  the  quadric  cone 
ax2  +  6^2  -f.  0^2  =  0  passes  through  the  vertex. 

6.  Prove  that  the  locus  of  the  point  of  intersection  of  three  mutually  per- 
pendicular tangent  planes  to  the  central  quadric  ax^  -f-  hy'^  ■\-  cz^  =\  is  the 

concentric  sphere  z^  +  y^  +  z'^  =  --\ 1 This  sphere  is  called  the  director 

a     b      c 

sphere  of  the  given  central  quadric. 

7.  Prove  that  through  any  point  in  space  pass  six  normals  to  a  given 
central  quadric,  and  five  normals  to  a  given  paraboloid. 

79.  Rectilinear  generators.  The  equation  of  the  hyperboloid 
of  one  sheet 


or 


or  also 


X'     y^      ^  —  1 

tie  form 

a      cj\a      CJ     \        b^ 

iH 

a     c            b 

1  +  ^'  ?_?' 
b      a     c 

a     c            0 

-  >. 

i-.y    ^_? 

b      a     c 

(14) 


(15) 


Let  the  value  of  each  member  in  (14)  be  denoted  by  ^,  so  that 
by  clearing  of  fractions  we  have 

For  each  value  of  ^,  these  equations  define  a  line.     Every  point 
on  such  a  line  lies  on  the  surface,  since  its  coordinates  satisfy 


94       PROPERTIES  OF  QUADRIC  SURFACES     [Chap.  VIII. 

(14).  Moreover,  through  each  point  of  the  surface  passes  a  line 
of  the  system  (16)  since  the  coordinates  of  each  point  on  the  sur- 
face satisfy  (14)  and  consequently  satisfy  (16).  The  system  of 
lines  (16),  in  which  |  is  the  parameter,  is  called  a  regulus  of  lines 
on  the  hyperboloid.  Any  line  of  the  regulus  is  called  a  generator. 
Similarly,  by  equating  each  member  of  (15)  to  yj,  we  obtain  the 
system  of  lines  whose  equations  are 

in  which  -q  is  the  parameter.  This  system  of  lines  constitutes  a 
second  regulus  lying  on  the  surface.  The  two  reguli  will  be 
called  the  ^  regulus  and  the  rj  regulus,  respectively.  Through 
every  point  P  of  the  surface  passes  one,  and  but  one,  generator 
belonging  to  each  regulus.  Moreover,  any  plane  that  contains  a 
generator  of  one  regulus  contains  a  generator  of  the  other  regulus 
also.  The  equation  of  any  plane  through  a  genei'ator  of  the  ^ 
regulus,  for  example,  may  be  written  in  tlie  form  (Art.  24) 


a      c        \        0 


bj        \a     c 


Since  this  equation  may  also  be  written  in  the  form 

it  follows  that  this  plane  also  passes  through  a  generator  of  the 
rj  regulus.  Every  such  plane  is  tangent  to  the  surface  at  the 
point  of  intersection  P  of  the  generators  in  it,  since  every  line 
in  the  plane  through  P  has  its  two  intersections  with  the  surface 
coincident  at  P. 

Example.     The  equations  of  the  reguli  on  the  hyperboloid 


4    '    9 

4-   ^  —  t  I   1 


-     +  ^   _  ^2  =  1 


X 

o   +   'S 


and  |+,  =  ,(,_!),     1+|  =  , (!-»)• 

The  point  (2,  6,  2)  lies  on  the  surface.     The  values  of  |  and  tj  which 


Arts.  79,  80]  ASYMPTOTIC  CONE  95 

determine  the  generators  through  this  point  are  |  =  1,  tj  =—  3.     Hence,  the 
equations  of  these  generators  are 

?  +  ^  =  l  +  2/,    i_y  =  ?_^,  and  ?  +  0=-3fl-2/V    i  +  ^=:_3f?-^V 
2  3'  3      2        '  2  V        3^  3  V2        / 

The  equation  of  the  plane  determined  by  these  lines  is 

This  is  the  equation  of  the  tangent  plane  at  (2,  6,  2)  (Art.  76). 
It  is  similarly  seen  that  the  equation 

a?     ¥ 
of  the  hyperbolic  paraboloid  may  also  be  written  in  the  forms 

ah         1         (. 


and 


Hence,  on  this  surface  also,  there  is  a  ^regulus  and  an  yj  regains 
The  generators  of  the  ^  regains  are  parallel  to  the  fixed  plane 

—  "  =  0 ;    those  of  the  -q  regains,  to  the  fixed  plane  "  +  -^  =  0. 
a      b  a      b 

By  writing  the  above  equations  in  homogeneous  coordinates,  it  is 

seen  that  the  line  -  -f  ^  =  0,  f  =  0  in  the  plane  at  infinity  belongs 
a      b 

to  the  $  regulus  ;  and  the  line  '  —  ^  =  0,  <  =  0  to  the  n  regains. 

a      b 

Hence  the  plane  at  infinity  is  tangent  to  the  paraboloid. 

The  hyperboloid  of  one  sheet  and  the  hyperbolic  paraboloid  are 

sometimes  called  ruled  quadrics,  since  the  reguli  on  them  are  real. 

It  will  be  shown  (Art.  115),  that  on  every  non-singular  quadric 

there  are  two  reguli ;  bat,  on  all  the  quadrics  except  these  two, 

the  reguli  are  imaginary. 

80.    Asymptotic  cone.     The  cone  whose  vertex  is  the  center  of 
a  given  central  quadric,  and  which  contains  the  curve  in  which 


2nz  ' 

X      y 

a      b 

X  _y 

a      b 

1 

2nz 

x^y 

a      b 

96  PROPERTIES  OF  QUADRIC  SURFACES     [Chap.  VIII. 

the  quadric  intersects  the  plane  at  infinity,  is  called  the  asymp- 
totic cone  of  the  given  quadric. 
If  the  equation  of  the  quadric  is 

the  equation  of  its  asymptotic  cone  is 

ax^  +  by"^  +  cz^  =  0. 

For,  this  equation  is  the  equation  of  a  cone  with  vertex  at  the 
center  (0,  0,  0,  1)  of  the  given  quadric  (Art.  46).  Its  curve  of 
intersection  with  the  plane  at  infinity  coincides  with  the  curve  of 
intersection 

of  the  given  surface  with  that  plane. 

EXERCISES 

(1.   Show  that  the  quadric  xy  =  z  \s  ruled.     Find  the  equations  of  its  gen- 
erators. 

2.  Show  that  x^  — 2 z^  +  by  —  x  +  9iz  =  0  is  Si  ruled  quadric. 

3.  Prove  that,  for  all  values  of  k,  the  line  x  +  \  =  ky  —  —  {k  +  \)z  lies  on 
the  surface  yz  +  zx  +  xy  +  y  +  z  =  0. 

4.  Prove  that  (y  +  rnz)  (x  +  nz)  —  z  represents  an  hyperbolic  paraboloid 
which  contains  the  X-axis  and  the  F-axis. 

5.  Show  that  every  generator  of  the  asymptotic  cone  of  a  central  quadric 
is  tangent  to  the  surface  at  infinity.  From  this  property  derive  a  definition 
of  an  asymptotic  cone. 

6.  Show  that  every  generator  of  the  asymptotic  cone  of  an  hyperboloid  of 
one  sheet  is  parallel  to  a  generator  of  each  regulus  on  the  surface. 

81.    Plane  sections  of  quadrics. 

Theorem  I.  The  section  of  a  quadric  by  a  finite  plane,  lohich  is 
not  a  component  of  the  surface,  is  a  conic. 

For,  let  TT  be  any  given  finite  plane,  and  let  the  axes  be  chosen 
so  that  the  equation  of  this  plane  is  2  =  0.  Let  the  equation  of 
the  quadric,  referred  to  this  system  of  axes,  be 

ax^+by^  +  cz''+2fyz+2gzx+2hxy+2lx+2my-^2nz  +  d=0.  (17) 
If,  when  2  =  0,  (17)  vanishes  identically,  the  given  quadric  is 


Abt.  80]  PLANE   SECTIONS  OF   QUADRICS  97 

composite  and  z  =  0  is  one  component ;  otherwise,  the  locus 
defined  in  the  XF-plane  by  putting  2;  =  0  in  (17)  is  a  conic. 

Theorem  II.  Tlie  sections  of  a  quadric  by  a  system  of  parallel 
planes  are  similar  conies  and  similarly  placed. 

Let  the  axes  be  chosen  so  that  the  equations  of  the  given  sys- 
tem of  parallel  planes  is  2;  =  A;,  and  let  (17)  be  the  equation  of  the 
given  quadric.     The  equation  of  the  projecting  cylinder  of  the 
section  by  the  plane  z  =  k  is 
ax^  +  '2hxy+by^  +  2(l+  gk)x  +  2  (?n  +fk)y  +  ck^  +  2nk  +  d  =  0. 

The  curves  in  which  these  cylinders  intersect  z  =  0,  and  conse- 
quently (Art.  45)  the  curves  of  which  they  are  the  projections, 
are  similar  and  similarly  placed,  since  the  coefficients  of  x'^,  xy, 
and  y"^  in  the  above  equation  are  independent  of  k* 

The  equations  of  the  section  of  the  surface  by  the  plane  at 
infinity  are  found  by  making  (17)  homogeneous  in  .-r,  y,  z,  t  and  put- 
ting t=  0.     They  are 

ax^  -\-  by^  +  cz"^  +  2  fyz +2  gzx -\- 2  hxy  =0,t  =  0. 

The  locus  of  these  equations  is  called  the  infinitely  distant  conic 
of  the  quadric.  This  conic  consists  of  two  lines  if  the  first  mem- 
ber of  the  first  equation  is  the  product  of  two  linear  factors.  The 
condition  for  factorability  is 

D  =  0. 


'iOfi 


EXERCISES 

lid  the  semi-axes  of  the  ellipse  in  which  the  plane  z  =  1  intersects 
the  quadric  x^  +  i  y"^  —  3  z^  +  i  yz  —  2  x  —  4  y  =  I. 

2.;  Show  that   the  planes  z  =  k  intersect  the  quadric  2  x'^  —  y"^  +  3  z^  + 
4  oiz^  2  yz  +  i  X  +  2  y  =  0  in  hyperbolas.     Find  the  equations  of  the  locus  of 
the-cgnters  of  these  hyperbolas. 
A    ( 3.   Show  that  the  curve  of  intersection  of  the  sphere  x"^  +  y"^  +  z"^  =  r^  and 

f\     ^ — ''  3.2         ,/2        g2 

the  ellipsoid  — f-  ^  +  —  =  1  lies  on  the  cone 
a2     b^     c^ 

a-z      r^j      ^\fy2      r"-)  W      r^ 

Find  the  values  of  r  for  which  this  cone  is  composite.     Show  that  each  com- 
ponent of  the  composite  cones  intersects  the  ellipsoid  in  a  circle. 

*  Cf.  Salmon,  "  Conic  Sections,"  6th  edition,  p.  222. 


98  PROPERTIES  OF  QUADRIC   SURFACES      [Chap.  VIII. 

82.    Circular  sections.     We  shall  prove  the  following  theorem  : 

Theorem  I.  Through  each  real,  finite  point  in  space  pass  six 
planes  which  intersect  a  given  non-composite,  non-spherical  quadric 
in  circles.  If  this  quadric  is  not  a  surface  of  revolution  nor  a  para- 
bolic cylinder,  these  six  p)lanes  are  distinct;  two  are  real  and  four 
are  imaginary.  If  the  quadric  is  a  surface  of  revohdion  or  a  para- 
bolic cylinder,  four  of  the  planes  are  real  and  coincident  and  two  are 
imaginary. 

Two  proofs  will  be  given,  based  on  different  principles. 

Proof  I.  Since  parallel  sections  of  a  quadric  are  similar,  it 
will  suffice  if  we  prove  this  theorem  for  planes  through  the  origin. 
The  planes  through  any  other  point,  parallel  to  the  planes  of  the 
circular  sections  through  the  origin,  also  intersect  the  quadric  in 
circles. 

Let  the  axes  be  chosen  in  such  a  way  that  the  equation  of  the 
quadric  is  (Art.  70) 

k.x''  -f-  W  +  ^-3^'  +  2  Ix  +  2  my  ^2nz  +  d  =  0,  (18) 

where  k^,  k^,  k^  are  the  roots  of  the  discriminating  cubic  (Art.  73). 
The  condition  that  a  plane  intersects  this  quadric  in  a  circle  is 
that  its  conies  of  intersection  with  the  given  quadric  and  with  a 
sphere  coincide. 

The  curve  of  intersection  of  the  quadric  (18)  with  the  sphere 

k(x~-\-y'^-^z'')-\-2lx-h2  7ny-\-2nz-^d  =  0  (19) 

coincides  with  the  intersection  of  either  of  these  surfaces  with 

the  cone 

(^•l  -  k)  x-"  +  {k^  -  k)y^  +  (k,  -  k)  z"  =  0. 

This  cone  is  composite  if  the  first  member  of  its  equation  is 
factorable,  that  is,  if  k  is  equal  to  k^,  k.,,  or  k^. 
It  follows  that  each  of  the  six  planes 


VA:i  -  k^  x=±  Vits  —kiy 
-Vki  —  ^2  X  =  ±  VA-2  —  k^z 
Vfcj  —  ki  y  =  ±  VA,*!  —  k^  z 

intersects  the  quadric  (18)  in  a  conic  which  lies  on  the  sphere  (19) 
and  is  consequently  a  circle. 


Art.  82]  CIRCULAR  SECTIONS  99 

If  A;,  >  k2  >  k^,  the  six  planes  are  distinct.     The  planes 
VA;i  —  fcj  a;  =  ±  VA-'j  —  fcj  2 
are  real.     The  others  are  imaginary. 

If  ki  =  k.2^k3,  the  last  four  planes  coincide  with  z  =  0.  The 
other  two  are  imaginary.  If  ki  =  ^2  ^  0,  the  quadric  (18)  is  a 
surface  of  revolution  (Art.  73).  If  k^  =  k.,  =  0,  it  is  a  parabolic 
cylinder  (Art.  To). 

If  the  equation  of  the  surface  is  in  the  form  (17),  and  k^,  k^,  k^ 
are  the  roots  of  its  discriminating  cubic,  it  follows  from  the  dis- 
cussion in  Article  73,  that  the  equations  of  the  planes  of  the 
circular  sections  through  the  origin  are 

ax2  +  bf  +  cz""  +  2fyz  +  2  gzx  +  2  hxy  -  k,  (x""  +  y^  +  z")  =  0, 
ax'^  +  by-  +  cz-  +  2fyz  +  2  gzx  +  2  hxy  —  k.-,  (x-^  +  y2-\-z'^)  =  0, 
ax2  +  6^2  ^  cz""  +  2fyz  +  2  gzx  +  2  hxy  -  ^-3  (a^  +  ?/'  +  2')  =  0. 

Proof  II.  It  was  shown  (Art.  49)  that  a  plane  section  of  a 
quadric  is  a  circle  if  it  passes  through  the  circular  points  of  its 
plane.  The  conic  in  which  the  quadric  meets  the  plane  at  infinity 
has  four  points  of  intersection  with  the  absolute.  Any  plane 
other  than  the  plane  at  infinity  which  passes  through  two  of 
these  points  will  meet  the  quadric  in  a  conic  through  the  circular 
points  of  the  plane ;  hence  the  section  is  a  circle. 

The  coordinates  of  the  points  of  intersection  may  be  found  by 
making  the  equations 

ax-  +  by"^  +  cz^  +  2fyz  +  2  gzx +  2  hxy  =  0,     x^  +  ?/2  +  z*  =  0 

simultaneous.  Since  both  equations  have  real  coefficients  and  the 
second  is  satisfied  by  no  real  values  of  the  variables,  it  follows 
that  the  four  points  Pj,  P.,,  P3,  P^  consist  of  two  pairs  of  conjugate 
imaginary  points,  or  of  one  pair  counted  twice. 

In  the  first  case,  let  Pj,  P^  be  one  pair  of  conjugate  points,  and 
P3,  P4  the  other.  The  lines  P^P.,  P^Pi  are  real  (Art.  41),  while  the 
lines  P1P3,  P2P4,  P1P4,  P2P3  are  imaginary.  The  pairs  of  lines 
PiPo,  P3P4;  P1P3,  P2P4;  PiP4>  P^Pz  constitute  composite  conies 
passing  through  all  four  of  the  points  Pj,  P2,  P3,  P,. 

In  the  second  case,  let  P.,  =  P4  and  Pj  =  P3.  The  lines  P1P2  and 
P3P4  coincide,  and  the  lines  PiP3,  P2P4  are  tangents  to  both  curves, 
which  have  double  contact  with  each  other  at  these  points. 


100        PROPERTIES  OF  QUADRIC  SURFACES     [Chap.  VIII. 

In  either  case  the  equations  of  the  lines  /*,Pt  can  be  found  as 
follows.  Through  the  points  of  intersection  of  (17)  and  the  abso- 
lute passes  a  system  of  conies 

ax'  +  b>f^cz-+2fiiz^2  gzx  +  2  hxi/  -  k{x--\-y^+z^)=0,  t=0.     (19') 

A  conic  of  this  system  will  consist  of  two  straight  lines  through 
the  four  points  of  intersection  if  its  equation  is  factorable,  that  is, 

a  —k        h  g    ' 

h        b-k        /     =0; 
9  f       c-k, 

thus  k  must  be  a  root  of  the  discriminating  cubic  (Art.  73).     Let 
A;,,  ki,  ks  be  the  roots  of  this  equation.     The  equations  of  the  pairs 
of  lines  are  then 
ax.2 4.  jjy2 ^ ez2 + 2 ./}/2  +  2  gz.v + 2  hxy -  k, {£■ + f  ^-7?)  =  ^,   t= 0,     (20) 

with  similar  expressions  for  ki  and  k^.  From  Art.  4t  it  follows 
that  for  one  of  the  roots  k^  the  two  factors  of  the  first  member  of 
the  quadratic  equation  (20)  are  real,  but  the  factors  for  each  of 
the  others  are  imaginary  when  the  roots  k^  are  all  distinct. 

If  ?«,  V  are  the  two  linear  factors  of  (20),  then  the  line  u  =  0, 
^  =  0  will  pass  through  one  pair  of  points  and  v  =  0,  <  =  0  will  pass 
through  the  other.  A  plane  of  the  pencil  u  +2?<  =  0  will  cut  the 
quadric  in  a  circle.  Since  a  plane  is  determined  by  a  line  and  a 
point  not  on  the  line,  the  theorem  follows. 

In  case  two  roots  of  the  discriminating  cubic  are  equal  and 
different  from  zero,  the  quadric  is  one  of  revolution;  the  two 
conies  in  the  plane  at  infinity  now  have  double  contact. 

If  fcj  >  k^  >  A'3,  the  planes  determined  by  the  second  root  are 
real. 

83.  Real  circles  on  types  of  quadrics.  The  above  results  will  now 
be  applied  to  the  consideration  of  the  real  planes  of  circular  section 
for  the  standard  forms  of  the  equation  of  the  quadric  (Chap.  VI). 

(a)  For  the  ellipsoid 

^'  +  ^  +  ^  =  1, 
a^      6^     (? 

the  roots  of  the  discriminating  cubic  are  1/a*,  1/6'',  1/c*. 


Art.  83]     REAL  CIRCLES   ON  TYPES   OF  QUADRICS        101 

Let  a  >  6  >  c  >  0.  Since  parallel  sections  of  the  surface  are 
similar,  it  follows  that  the  equations  of  the  real  planes  of  circular 
section  are 


cy/a''-¥x±a^b''-c^z-{-d  =  0,  (21) 

where  d  is  a  real  parameter. 

The  circle  in  which  a  plane  (21)  intersects  the  ellipsoid  is  real 
if  the  plane  intersects  the  ellipsoid  in  real  points,  that  is,  if  it  is 
not  more  distant  from  the  center  than  the  tangent  planes  parallel 
to  it.  The  condition  for  this  is  (Arts.  76  and  16)  |  d  ]  £.  ac^a?  —  fl 
If  I  d  I  >  ac  Va^  —  c^  the  circles  are  imaginary. 

If  1^1=  ac Va^  —  c",  the  circles  are  point  circles.  The  four 
planes  determined  by  these  two  values  of  d  are  the  tangent  planes 
to  the  ellipsoid  at  the  points 

-,  0,    ±cyA- 

'  ^a^  —  c 

Each  of  these  points  is  called  an  umbilic. 

The  two  systems  of  planes  (21)  are  also  the  real  planes  of  circu- 
lar section  of  the  imaginary  cone 

^4.^  +  5^=0, 
a?     b^     c^        ' 

and  of  the  imaginary  ellipsoid 

^!  4. 2/'  >  ?"  ^  _  1 
a^     b^     c" 

(b)  The  equations  of  the  real  planes  of  circular  section  of  the 
hyperboloids  of  one  and  two  sheets 


and  of  the  real  cone 


a""     b''     c2  ' 


a"      62      c2       ' 


where  a  >  6  >  0,  are  found  to  be 


c  Va2  -b^y  ±b  Va^  +  e"  z  +  d  =  0. 


102        PROPERTIES  OF  QUADRIC  SURFACES     [Chap.  VIII. 

On  the  hyperboloid  of  one  sheet  and  the  real  cone,  the  radii  of 
the  circles  are  real  for  all  values  of  d.  On  the  hyperboloid  of  two 
sheets,  the  circles  are  real  only  if  \d\  >bc  ^ b^  +  c^.  The  coordi- 
nates of  the  umbilics  on  the  hyperboloid  of  two  sheets  are 

(c)  The  real  planes  of  circular  section  of  the  elliptic  paraboloid 
'^  +  l  =  2nz,  a>6>0,  n>0 

and  the  real  or  imaginary  elliptic  cylinders 


-,  +  f  =±1,  a>6>0 


are  determined  by 


±  V«2  -  b'  y  +  bz  +  d  =  0. 


On  the  real  elliptic  cylinder,  the  circles  are  real,  and  on  the 
imaginary  cylinder  they  are  imaginary,  for  all  values  of  d.     On 

the  elliptic   paraboloid,  the  circles  are    real    if   d<-^(a^  —  b"^). 

The  coordinates  of  the  umbilics  on  the  elliptic  paraboloid  are 

n 


0,    ±bn^(e--b-,   ^{a'-b') 

(d)  For  the  hyperbolic  paraboloid 

2nzt 


x^_y^_r, 


a"     b' 
and  the  hyperbolic  cylinder 

^2  ^  y2  ^  ^^ 

a"     b' 
the  equations  of  the  planes  of  the  circular  sections  are 

bx  ±  ay  -\-  dt  =  0. 
The  circles  in  these  planes  are  all  composite.     For,  the  planes 

bx  +  ay  -\-  dt  =  0 


Art.  83]       REAL  CIRCLES   ON   TYPES   OF   QUADRICS       103 

intersect  these  surfaces  in  the  fixed  infinitely  distant  line 

hx  +  ay  —  0,   t  =  0 

and  in  a  rectilinear  generator  which  varies  with  d.     Similarly, 

the  planes 

hx  —  ay  -\-dt  =  0 

intersect  them  in  the  line 

hx  —  ay  =  0,    t  =  0 

and  in  a  variable  generator. 
Also  on  the  parabolic  cylinder 

a;2  =  2  myt 

the  real  circles  are  all  composite,  since  the  planes  x  =  dt  intersect 
the  surface  in  the  fixed  line  x  =  t=  0,  and  in  a  variable  generator. 
We  have,  therefore,  the  following  theorem  : 

Theorem  II.  On  the  hyjjerhoh'c  2^("'<^l>oloid,  the  hyperholic 
cylinder,  and  the  j^'^i'f'ci'hoUc  cylinder,  the  real  circular  sections  are 
composite.  The  components  of  each  circle  are  an  infinitely  distant 
line  and  a  rectilinear  generator  ivhich  i)itersects  it. 


^ 


EXERCISES 


f  1.:  Find    the   equations   of    the    real   circular    sectipns   of    the   surface 
4  '^^  2  2/2  +  z^  +  3yz  +  X2  =  1.    ^>H^^  '^  "^  "^^  P*^  -kCit^-^'i  "^^  J*)  -'  «  «> 

2.,  Find   the    equations   of    the   real   circular   sections    of    the    surface 
2  x^+  5  2/2  +  3  5;-  +  4  .r2/  =  1. 
i'  3.   Find  the  radius  of  a  circular  section  through  the  origin  in  Ex.  2. 
.-^.!  Find  the  equations  of  the  real  planes  through  (1.  —3,  2)  which  in- 
tetsfect  the  ellipsoid  2  x-  +  y'^  +  iz'^  =  1  in  circles. 

5.   Find  the  conditions  which  must  be  satisfied  by  the  coefficients  of  the 
equation  F{x,  y,  z)  =  0  oi  a,  quadric  if  the  planes  z  =  k  intersect  it  In  circles. 

6.'   Show  that  the  centers  of  the  circles  in  Ex.  5  lie  on  a  line.     Find  the 
equations  of  this  line. 

//y.    Find  the  second  .system  of  real  planes  cutting  circles  from  the  quadric 
Kn  Ex.  5. 

8.  Find  the  conditions  which  must  be  satisfied  by  the  coefficients  if  the 
plane  Ax  +  By  +  Cz  -{-  D  =  0  intersects  the  quadric  F{x,  y,  z)  =  0  in  circles. 

9.  Find  the  coordinates  of  the  center  and   the   radius   of  the  circle  in 
which  the  plane  x  =2  z  +  5  intersects  the  cone  3  x-  +  2y'^  —  2  z^  =  0. 


^tp 


±-i 


45^^ROPERTIES  OF  QUADRIC  SURFACES     [Chap.  VIII. 


V    10.    Show  that,  for  all  values  of  \,  the  equation  of  the  planes  of  the  cir- 
cular sections  of  the  quadrics 

(a  +  X)x2  +  (6  4-  X)2/2  +  (c  +  \)z2  =  1 

are  the  same.     The  quadrics  of  this  system  are  said  to  be  concyclic. 

84*    Confocal  quadrics.     The  system  of  surfaces  represented  by 
the  equation 


+  -^—  +  -^^—  =  1,  a>b>c>0,  (22) 

in  which  k  is  a  parameter,  is  called  a  system  of  confocal  quadrics. 
The  sections  of  the  quadrics  of  the  system  by  the  principal  planes 
x  =  0,  y  =  0,  z  =  0  are  confocal  conies. 

If  A;  >  —  c^,  the  surface  (22)  is  an  ellipsoid ;  if  —  c-  >k>  —  b"^, 
the  surface  is  an  hyperboloid  of  one  sheet;  if  —  6-  >  fc  >  —  a^,  the 
surface  is  an  hyperboloid  of  two  sheets ;  if  —  a^  >  k,  the  surface 
is  an  imaginary  ellipsoid. 

If  A:>  — c^,  but  approaches  —  c^  as  a  limit,  the  minor  axis  of 
the  ellipsoid  approaches  zero  as  a  limit,  and  the  ellipsoid  ap- 
proaches as  a  limit  the  part  of  the  XF-plane  within  the  ellipse 

-^—  +  -^^  =  1.  (23) 

If  —  e^>A;  >  —  b"^,  the  surface  is  an  hyperboloid  of  one  sheet. 
As  k  approaches  —  c-,  the  surface  approaches  the  part  of  the 
XF-plane  exterior  to  the  ellipse  (23).  As  A;  approaches  —  6^,  the 
surface  approaches  that  part  of  the  XZ-plane  which  contains  the 
origin  and  is  bounded  by  the  hyperbola 

=  1.  (24) 


a^  —  b"^     b^  —  c^ 

If  —  ¥  >k>  —  o},  the  surface  is  an  hyperboloid  of  two  sheets. 
As  k  approaches  —  6^,  the  hyperboloid  approaches  that  part  of  the 
plane  y  =  0  which  does  not  contain  the  origin.  As  k  approaches 
—  ci^,  the  real  part  of  the  surface  approaches  the  plane  a;=0, 
counted  twice. 

The  ellipse  (23)  in  the  XF-plane  and  the  hyperbola  (24)  in  the 
XZ-plane  are  called  the  focal  conies  of  the  system  (22). 


Arts.  84,  85]  ELLIPTIC  COORDINATES  105 

The  vertices  of  the  focal  ellipse  are 


(-t-Va^-cS  0,  0). 
The  foci  are 

(±Va^^^^  0,  0). 

On  the  focal  hyperbola  the  vertices  are  ( ±  Va^  —  6^  0,  0)  and  the 
foci  are  (±  Va^  — c^,  0,  0).  Hence,  on  the  focal  conies,  the  ver- 
tices of  each  are  the  foci  of  the  other. 

85.   Confocal  quadrics  through  a  point.     Elliptic  coordinates. 

Theorem  I.  Three  confocal  quadrics  pass  through  every  point 
P  in  space.  If  P  is  real,  one  of  these  quadrics  is  an  ellipsoid,  one  an 
hyperholoid  of  one  sheet,  and  the  third  an  hyperholoid  of  two  sheets. 

If  P  =  (Xi,  yi,  Zi)  lies  on  a  quadric  of  the  system  (22),  the  param- 
eter k  satisfies  the  equation 

{j£  +  a''){k  +  b'){k  +  c^) -  x^(k  -f  ¥)(k  +  c^)  -  y,\k  +  c%k  +  a'-) 

-  z,\k  +  a^k  -f  h^)=  O.sr^i  -1^X25) 

Since  this  is  a  cubic  equation  in  k,  and  each  of  its  roots  determines  .  9- 
a  quadric  of  the  system  through  P,  there  are  three  quadrics  of  eT  *^ 
the  system  (22)  which  pass  through  P.  ^'jJc 

Let  P  be  real.  **«> 

If  ^'  =  -f-  00,  the  first  member  of  (25)  becomes  positive.     -Mj  h    1  / 

If  A;  =  —  c^,  it  is  —  2;i2(— c2-|-a2)(— c^  4- &2)^-vvhich  is  negative.  =V^'^i 
If  k  =  —  b"^,  it  is  —  yi'^(  —  ¥  +  c'^){—  ¥  +  a-),  wliich  is  positive.   ^  -f  C~ ^^< 
Itk  =  —  a^,  it  is  —  x^%  —  a^  +  ?/)( —  a^  +  c^),  which  is  negative.   ^  ■}  '  ~* 

Hence  the  roots  of  (25)  are  real.     One  is  greater  than  —  c^,  one 
lies  between  —  c^  and  —b^,  and  the  third  between  —  6^  and  —  a^- 
Denote  these  roots  by  ki,  k^,  k^.     Hence,  we  have 

ki  >-c'>k,>-¥>k,>-  a\ 

Then,  of  the  three  quadrics 

f 


1,9!      I       7.         '        _9      1       7.  ' 


a2  -\-k,      b''  +  k,      c2  -1-  fci 

CC^  iP"  2*^ 


106        PROPERTIES  OF  QUADRIC  SURFACES 


rj 

[Chap.  VIII.  ^^ 


which  pass  through  P,  the  first  is  an  ellipsoid,  the  second  an  hy- 
berboloid  of  one  sheet,  and  the  third  an  hyperboloid  of  two  sheets. 

Theorem  II.     TJie  three  quadrics  of  a   confocal  system  which 
pass  through  P  intersect  each  other  at  right  angles. 

For,  the  equations  of  the  tangent  planes  to  the  first  two  quad- 
rics (26)  are 


^ 


^ 


a^  +  K 


+ 


4- 


62  +  A;i 


+ 


+ 


(?  +  fci 


a^  +  k.  '  5-  + A-.,  '  c'^  +  k^ 
These  planes  are  at  right  angles  if 


=  1, 
=  1. 


V 

n 

t 

S 


'S^ 

<'.'? 


+ 


Vi 


+  ■ 


(a^  +  ;ci)(a2  +  k,)     {1/  +  k,){b'  +  k.^      (c^  +  k,){c''  +  k,) 


=  0 


That  this  condition  is  satis- 
fied is  seen  by  substituting  the  Jj-*] 
coordinates  of  P  in  (26),   sub*!*" 
tracting    the    second    equation? 
from  thefirst,  and  removing  the*^' 
factor  A'2— A'l,   which  was  seen 
to  be  different  from  zero.     The 
proof  for  the  other  pairs  may 
be  obtained  in  the  same  way. 

The  three  roots  k^,  k^,  A3  of 
equation  (25)  are  called  the  el- 
liptic coordinates  of  the  point  P. 
To  find  the  expressions  for  the 
rectangular  coordinates  of  P,in 
terms  of  the  elliptic  coordinates, 
we  substitute  the  coordinates 
(.Ti,  ?/i,  2,)  of  P  in  (26)  and  solve 
for  Xi^,  yx,  %{-.     The  result  is 


,.,_(«^  +  A-0(a^  +  A,)(a^  +  A3) 
'  (a2-62)(a2-c2)         ' 

{WJrK^{W^-k^{W^k,^ 

^'  (^2  _  (j2)(52  _  c2)  ' 

,^(c''  +  A0(c^  +  A,)(c^-j-A-3) 
'  (c2-a2)(c2-62) 


(27) 


Arts.  85,  86]         QUADRICS  TANGENT  TO  A  LINE  107 

It  is  seen  at  once  from  these  equations  that  Ji\,  fcj,  and  ^'3  are  the 
elliptical  coordinates,  not  only  of  P,  but  also  of  the  points  sym- 
metric with  P  as  to  the  coordinate  planes,  axes,  and  origin. 

86.    Confocal  quadrics  tangent  to  a  line. 

Theore3i  I.     An.i/  line  touches  tivo  quadrics  of  a  confocal  system. 

The  points  of  intersection  of  a  given  line  with  a  quadric  of  the 

system  (22)  are  determined  by  the  equation  (Art.  65)  oL^  U^aaa^  ■*"*  3*< 

witC  *>  14/  i'v  Sam** 


a"  -\-k      ¥  +  k      c-  +  kj  \a''  +  k      b^  +  k      c^~  +  k 

W  +  k     b^  +  k     c-'  +  k       J  ^ 

The  condition  that  this  line  is  tangent  is  -^  y     j^-a..*:^  w^  vJ^^^ 

a^j^kh-'^kc'^k)  ^^"^ 


+  v;;^  +  -;r^     -^F^  +  t/^  +  ^t^-I    =0. 


When  expanded  and  simplified,  this  equation  reduces  to 

y?  -f  [(62  +  c2)  A^  +  {c?  +  «')  /x^  +  («'  +  /'^) v^  -  (a^oi^  -  y,\Y 

—  (z/o«'  —  ^Jq/a)"  —  (2;o^-  —  -I'd")']  ^  +  [ft-c^A^  +  c^aV  +  a-lP-v^ 

—  i?'^l>'  -  .'/o^)c'  -  (.Vol'  -  2;oa)/j-  —  (^oA.  -  .Tov)a2]  =  0. 

Since  this  equation  is  quadratic  in  k,  the  theorem  follows. 

Theorem  II.     If  tico  confocal  quadrics  touch  a  line,  the  tangent 
'A.  planes  at  the  points  of  contact  are  at  right  angles. 

fl       Let   A"i   and   A'o   be   the   parameters    of   the   quadrics,  and   let 

^     P'={x',  y',  z'),  P"  =  {x",  y",  z")  be  the  points  of  tangency  of  the 

\^^line   with   the  given    quadrics.     The   equations  of   the   tangent 

'.J  planes  at  P'  and  P"  are  (Art.  76),  respectively, 

jj  ;j»        x'x  y'y  ^'^     =\         ^"^      1     ?/"-^^     1     ^'''^    —\ 

,^      a^+A-i      h'^-^ki^  c'-^-k^        '     a2  +  A-o      &2  ^  A:.,      c2  +  A;2 

These  planes  are  at  right  angles,  if 

^'*^"  + ^NL + ^^ =  0.       (28) 


(a2  +  A'0(a2  +  A-o)      (6^  +  a-,)(62  +  A%)      (c^  +  A-i)(c2  +  A-,) 


I 


l08       properties   of  QUADRIC  surfaces      [Chap.  VIII. 

Since  the  line  through  P'  and  P"  is  tangent  to  both  quadrics,  it 
lies  in  the  tangent  planes  at  both  points.  Hence  P*  and  P"  lie  in 
both  planes,  so  that 

,.U"  ,.'-.,"  /v'-v"  ^'t"  ii'^/"  -^'-v" 


1 


J  u.^      ,    yy",     z-z"  ^^      x'xr         yy        z'z"  ^^  /,j> 

I*!     V  ,.■>    ,     1.        '      -LO    ,     1.        '        .9     I     7.  '        „9     1     7«        '"    t9    1     7.        '       ,.01     1     7. 


a2+A:j      62+^"j      c2  +  ;fci        '    o?  +  k^      b'  +  k.      c'+h        '  J^<' 

»       xjj  subtracting  one  of  these  equations  from  the  other,  it  is  seen';^^  ^ 
J       that  (28)  is  satisfied.     The  planes  are  therefore  at  right  angles.    ,     ^/ 


87.    Confocal  quadrics  in  plane  coordinates.    The  equation  of  tl^  ^/ 
system  (22)  in  homogeneous  plane  coordinates  (Art.  77)  is      /v^"'^     *^ 

ahr  +  6V  +  c'^iv^  -  s^  +  k(if  +  w^  +  w^)  =  0.  ■^y   *  V^  v* 

Since  this  equation  is  of  the  first  degree^i  k,  we  have  the  follow-   ^* 
ing  theorem:6jc^  (^''+  A.;!^,V^.  •        -  •    V      U;-^^  M^^'+'^V^iJl 

Theorem.  An  arbitrary  plane  («i,  Vj,  ?f'i,  s,)t^s  tangent  to  one 
and  onlij  one  quadric  of  a  confocal  system.         -  _  ^^t^-^^JUXc  Lc^*-n/^  "^ 

The  (imaginary)  planes  whose  homogeneous  coordinates  satisfy 
the  two  equaUons^frv^i_JQj_^^^^^^  r  *  J 

"a^M^  +  6^y^  +  c^zo'  —  s^  =  0,     w'^  +  V"  +  w^  =  0 
are  exceptional.      They  touch  all   the  quadrics  of   the  system. 
Hence,  all  the  quadrics  of  a  confocal  system  touch  all  the  planes 
common  to  the  quadric  k  =  0  and  the  absolute. 

EXERCISES 

*»  1.  Prove  that  the  difference  of  the  squares  of  the  perpendicular  from  the 
center  on  two  parallel  tangent  planes  to  two  given  confocal  quadrics  is  con- 
stant. This  may  be  used  as  a  definition  of  confocal  quadrics. 
^2.  Prove  that  the  locus  of  the  point  of  intersection  of  three  mutually  per- 
pendicular planes,  each  of  which  touches  one  of  three  given  confocal  quadrics, 
is  a  sphere. 

3.  Write  the  equation  of  a  quadric  of  the  system  (22)  in  elliptic  coordi- 
nates. Derive  from  (27)  a  set  of  parametric  equations  of  tliis  quadric,  using 
elliptic  coordinates  as  parameters.  r,     \ 

*(    4.   Discuss  the  system  of  confocal  paraboloids         y      i  V^  s?a  ^^^^ 


a^  +  k     b-^  +  k 
5.   Discuss  the  confocal  cones 


+  — ^ —  =  2iiz  +  kn^. 


i^vA 


+  -^ —  +  _^=0. 


a^  +  k      b'^  +  k      c-  +  k 


<AAZ-<iyiutr4  rxi 'y^'^^n^^^ 


CHAPTER    IX 

TETRAHEDRAL  COORDINATES 

88.  Definition  of  tetrahedral  coordinates.  It  was  pointed  out  in 
Art.  34  that  the  four  planes  x  =  0,  ?/  =0,  z  —  0,  and  t  =  0,  which 
do  not  all  pass  through  a  point,  may  be  considered  as  forming  a 
tetrahedron  which  was  called  the  coordinate  tetrahedron.  We 
shall  now  show  that  a  system  of  coordinates  may  be  set  up  in 
which  the  tetrahedron  determined  by  any  four  given  non-concur- 
Ter^t/  planes  is  the  coordinate  tetrahedron.  A  system  of  coordi- 
nates so  determined  will  be  called  a  system  of  tetrahedral 
coordinates. 

Let  the  equations  of  the  four  given  non-concurrent  planes  (re- 
ferred to  a  given  system  of  homogeneous  coordinates)  be 

AiX  +  B,y  +  C,z  +  D,t  =  0,         i=l,  2,  3,  4.         (1) 

Since  these  planes  do  not  all  pass  through  a  point,  the  determinant 


^1 

B, 

c. 

A 

A 

Bo 

c. 

A 

A 

Bs 

c, 

A 

A, 

B, 

c. 

A 

0  <^^^^^^fOjO^O^» 


does  not  vanish. 

Let  the  coordinates  {x,  y,  z,  t)  of  any  point  P  in  space  be  sub- 
stituted in  the  first  members  of  (1)  and  denote  the  values  of  the 
resulting  expressions  by  a-j,  X2,  x^,  x^,  respectively,  so  that 

x^  =  A^x  +  Bin  +  CiZ  +  D^t, 

x^  =  A,x  +  BsV  +  CsZ  +  D.,t,  ^  ^ 

Xi  =  AiX  +  B^y  +  C4Z  +  Dit. 

We  shall  call  the  four  numbers  x^,  x^,  x^,  x^  determined  by  these 
equations  the  tetrahedral  coordinates  of  P.  The  four  planes  (1)  are 
called  the  coordinate  planes.  Their  equations  in  tetrahedral  coor- 
dinates are  x^  =  0,  x.,  =  0,  x^  =  0,  and  Xi  =  0,  respectively. 

133 


110  TETRAHEDRAL   COORDINATES  [Chap.  IX. 

Since  the  four  planes  (1)  do  not  all  pass  through  a  point,  the 
coordinates  x^,  X2,  x^,  x^  cannot  all  be  zero  for  any  point  in  space. 
When  (x,  ?/,  z,  t)  are  given,  the  values  of  a-j,  x.,,  x^,  x^  are  uniquely 
determined  by  (3).  Conversely,  since  the  determinant  (2)  does 
not  vanish,  equations  (3)  can  be  solved  for  x,  y,  z,  t  so  that,  when 
Xi,  X2,  X3,  x^  are  given,  one  and  only  one  set  of  values  of  x,  y,  z,  t 
can  be  found.  Since  (.r,  y,  z,  t)  and  (kx,  ky,  kz,  kt)  represent  the 
same  point  (Art.  29),  it  follows  from  (3)  that  (x^,  x^,  x^,  x^  and 
{kx^,  kx-i,  kx^,  kx^)  represent  the  same  point,  k  being  an  arbitrary 
constant,  different  from  zero. 

89.  Unit  point.  A  system  of  tetrahedral  coordinates  is  not 
completely  determined  when  the  positions  of  its  coordinate  planes 
are  known.     For,  since  the  equations 

k(Ax  +  By  -{-Cz  +  Dt)  =  0,         k^  0, 
and  Ax  +  By  -\-Cz-\-Dt  =  0 

represent  the  same  plane  (Art.  24),  it  follows  that  if  k^,  ko,  k^,  k^ 
are  four  arbitrary  constants  differejit  from  zero,  the  equations 

x',  =  k^(A,x  +  B,y  +  C\z  +  D,t),         i  =  1,  2,  3,  4  (4) 

define  a  system  of  tetrahedral  coordinates  having  the  same  coordi- 
nate planes  as  (3)  but  such  that 

x\  =  k-x-,  i  =  1,  2,  3,  4. 

The  point  whose  tetrahedral  coordinates  with  respect  to  a  given 
system  are  all  equal,  so  that  x^:  x^:  x^:  Xi  =  l :  1 : 1  -.1,  is  called  the 
unit  point  of  the  system. 

Theorem  I.  Any  j)oint  P,  not  lying  on  a  face  of  the  coordinate 
tetrahedron,  may  he  taken  as  unit  point. 

For,  by  substituting  the  coordinates  (x,  y,  z,  t)  of  P  in  (4) 
values  of  k^,  k.^,  k^,  ki  may  be  found  such  that  «/  =  x^  '=  .^3'  =  x/^ 
so  that  P  is  the  unit  point. 

Since  the  ratios  A:, :  ^2 :  ^3 :  A:4  are  fixed  when  the  unit  point  has 
been  chosen,  we  have  the  following  theorem  : 

Theorem  II.  The  system  of  tetrahedral  coordinates  is  deter- 
mined xvhen  the  coordinate  planes  .r,  =  0,  .Tj  =  0,  cCg  =  0,  a^^  =  0  and 
the  unit  point  (1,  1,  1,  1)  liave  been  chosen. 


Arts.  89,  90]  EQUATION   OF  A  PLANE  HI 


EXERCISES 

In  the  following  problems,  the  equations  in  homogeneous  coordinates  of 
the  coordinate  planes  of  the  given  system  of  tetrahedral  coordinates  are 

»  X  -  y^+2t  =  0,  x+2y  —  2z-{-t  =  0,      --'^ 

3x  +  3y  +  2z  +  2t-0,  x-Sy  +  z  +  2t=0. 

The  homogeneous  coordinates  of  the  unit  point  are  (—  1,  2,  —  1,  1). 

1.  Find  the  tetrahedral  coordinates  of  the  points  whose  homogeneous 
rectangular  coordinates  are  (x,  y,  z,  t),  (0,  0,  0,  1),  (1,  1,  1,  1),  (5,  1,  —  2,  1), 
(8,  1,1,0),  (0,1,  -1,0). 

2.  Find  the  rectangular  coordinates  of  the  points  whose  tetrahedral 
coordinates  are  (—  1,  1,  4,  3),  (1,  2,  —  1,  —  5),  (0,  0,  1,  3),  (xi,  Xi,  x^,  X4). 

3.  Write  the  equation  of  the  surface  xi  +  2^2  —  2  X3  —  a;4  =  0  in  rec- 
tangular coordinates.     Show  that  the  locus  is  a  plane. 

4.  Write  the  equation  of  the  plane  [>  x  +  y  +  z  —  t  =  0  in  tetrahedral 
coordinates. 

r^       5.   Write   the    equation   of   the    surface    XiXo  +  x^Xi  =  0    in   rectangular 

I  ^j  coordinates. 

♦^    ■ . — 

^  6.   Solve  Exs.  1  and  2  when  the  point  whose  rectangular  coordinates  are 

^Q  (3,  1,  —  2,  2)  is  taken  as  unit  point. 


S^ 


7.    Why  may  not  anoint  lying  in  a  face  of  the  tetrahedron  of  reference 
(      be  taken  as  unit  point  ./to i  ^^O^.^,^,^^^,^^^    ^e> 

,  f  90.  Equation  of  a  plane.  Plane  coordinates.  From  the  equation 
/[^  Kx  -f  vy  -f-  wz  -\-  sf  =  0  (5) 

y»  of  a  plane  in  homogeneous  rectangular  coordinates,  the  corre- 
^  spouding  equations  in  tetrahedral  coordinates  can  be  found  by 
"5(5^  solving  equations  (3)  for  x,  y,  z,  t  and  substituting  in  (5).  The 
^^  resulting  equation  is  linear  and  homogeneous  in  Xi,  x.2,  x^,  x^  of 
:!^     the  form 

^  Wj-Tl  +  U^X^  +  UsXs  +  U^Xi  =  0,  (6) 

J*       .  . 

*      with  constant  coefficients  ^^■^,  ti^,  113,  W4.     Conversely,  any  equation 

T     of  the  form  (6)  defines  a  plane.     For,  if  x^,  x^,  x^,  X4  are  replaced 

4    by  their  values  from  (3),  the  resulting  equation  is 
I* 

ux  +  vy-^  wz  -f-  si  =  0, 


112  TETRAHEDRAL  COORDINATES  Chap.  IX. 


wherein  xi,  =  A^i^  +  Am^  +  A^u^  -\-  -44M4, 

V  =  A«i  +  B^iL,  +  B^u^  +  54W4, 
ty  =  Ci^i  +  (72?t2  +  (73H3  +  (74?/4, 


(7) 


The  coefficients  u^,  Uo,  u^,  u^  in  (6)  are  called  the  tetrahedral 
coordinates  of  the  plane  (compare  Arts.  27  and  29).  It  follows 
from  equations  (7)  and  (2)  that,  if  u^,  U2,  x(s,  '«4  (not  all  zero)  are 
given,  the  plane  is  definitely  determined,  and  that,  if  the  plane  is 
given,  its  tetrahedral  coordinates  (wj,  U2,  ih,  it^  are  fixed  except  for 
an  arbitrary  multiplier,  different  from  zero. 

91.  Equation  of  a  point.  Let  {x^,  x^,  x^,  x^)  be  the  coordinates 
of  a  given  point.  The  condition  that  a  plane  whose  coordinates 
are  (u^,  xu,  7/3,  xi^)  passes  through  the  given  point  is,  from  (6) 

Wl^i  +  Xl..^2  +  ^'3^'3  +  "4»"4  =  0.  (8) 

This  equation,  which  is  satisfied  only  by  the  coordinates  of  the 
planes  which  pass  through  the  given  point,  is  called  the  equation 
of  the  point  (xj,  x^,  x^,  x^)  in  plane  coordinates  (cf.  Art.  28). 

It  should  be  noticed  that,  in  the  equation  (6)  of  a  plane, 
(?fj,  U2,  M3,  W4)  are  constants  and  (x^,  Xo,  x^,  Xi)  are  variables.  In 
the  equation  (8)  of  a  point  (x^,  x.,,  Xg,  X4)  are  constants  and 
(wj,  U2,  u-i,  W4)  are  variables. 

92.  Equations  of  a  line.  The  locus  of  the  points  whose  coordi- 
nates satisfy  two  simultaneous  linear  equations 

U'\X^  +  ?/"2iC2  +  U'^X^  +  ^"4X4  =0  ^    ^ 

is  a  line  (Art.  17).  The  two  simultaneous  equations  are  called 
the  equations  of  the  line  in  point  coordinates. 

Similarly,  the  locus  of  the  planes  whose  coordinates  satisfy 
two  simultaneous  linear  equations 

X  jMi  +  X  2?^2  "T  ^  3^*3  "I    "^  4^*4  ^^  ")  /-I  A\ 

JC'>,  +  CC"2W2  +  •'C'V'3  +  a^'>4  =  ^ 

is  a  line  (Art.  28).  These  two  simultaneous  equations  are  called 
the  equations  of  the  line  in  plane  coordinates. 


Arts.  91-93]  DUALITY  113 

EXERCISES 

1.  Write  the  equations  and  the  coordinates  of  the  vertices  and  of  the 
faces  of  the  coordinate  tetrahedron. 

2.  Write  the  equations  in  point  and  in  plane  coordinates  of  the  edges  of 
the  coordinate  tetrahedron. 

3.  Find  the  equations  of  the  folio  wing  points:  (1,  1,  1,  1),  (3,  —  5,7,  —  1), 
(_  1,  6,  -4,  2),  (7,  2,  4,  6).  ,.r 

4.  Write  the  coordinates  of  the  following  planes  : 

^\  +  y-i.  +  X3  +  a;4  =  0,  7  .Ti  —  X2  —  3  .rs  ^  X4  =  0,  a;i  +  9  j-2  —  5  X3  —  2  X4  =  0. 

5.  Write  the  equations  of  the  line  Xi  +  0:2  =  0,  X3  —  7  3:4  =  0  in  plane 
coordinates. 

ScG.     Write  the  equations  of  two  points  on  the  line. 

6.  Find  the  coordinates  of  the  point  of  intersection  of  the  planes  (1,  2,  7, 
3),  (1,  3,  6,  0),  (1,  4,  5,  2). 

93.  Duality.  We  have  seeu  that  any  four  numbers  x^,  X2,  x^,  x^, 
not  all  zero,  are  the  coordinates  of  a  point  and  that  any  four  num- 
bers ?<i,  U2,  U3,  ?<4,  not.  all  zero,  are  the  coordinates  of  a  plane. 
The  condition  that  the  point  (xy,  x^,  x^,  x^)  lies  in  the  plane 
(«,,  1*2,  Uz,  Ui),  or  that  the  plane  (11^,  U2,  u^,  u^  passes  through  the 
point  (Xi,  X2,  x^,  x^  is 

U^Xi  +  ?<2^*2  +  ^'3^3  +  ^<4^4  =  0. 

This  equation  remains  unchanged  if  x^,  x^,  x^,  x^  and  u^,  u^,  W3,  u^ 
are  interchanged. 

The  equations  (9)  and  (10)  of  a  line  are  simply  interchanged  if 
point  and  plane  coordinates  are  interchanged. 

From  the  above  observations,  the  following  important  principle, 
called  the  principle  of  duality,  may  be  deduced ;  namely,  that  if 
we  interchange  Xj,  x^,  x^,  x^  and  u^,  u^,  U3,  u^  in  the  proof  of  a 
theorem  concerning  the  incidence  of  points,  lines,  and  planes,  or 
concerning  point  and  plane  coordinates,  we  obtain  at  once  the 
proof  of  a  second  theorem.  The  theorem  so  derived  is  called  the 
dual  of  the  first.  It  is  obtained  from  the  given  one  by  inter- 
changing the  words  point  and  plane  in  the  statement. 

In  the  next  two  Articles  we  shall  write  side  by  side  for  com- 
parison the  proofs  of  several  theorems  and  their  duals. 

The  symbols  (x),  (x'),  (u),  etc.,  will  be  used  as  abbreviations  for 
(xi,  Xj,  Xj,  X4),  (x\,  x'2,  x'3,  x\),  (itj,  U2,  M3,  iCi)  etc.,  respectively. 


114 


TETRAHEDRAL   COORDINATES  [Chap.  IX. 


94.    Parametric  equations  of  a  plane  and  of  a  point. 


Xi 

1^2 

.T3 

X, 

x\ 

dC  2 

^•'3 

x 

x'\ 

^\ 

x'\ 

X 

»h       2 

x"\ 

X 

Let  {x'),  (x"),  (x'")  be  three 
given  non-collinear  points.  Tlie 
equation  of  tlie  plane  detei"- 
mined  by  them  is  found,  by 
the  same  metliod  as  that  em- 
ployed in  Art.  11,  to  be 


=  0.  (11) 


Let  (.r)  be  any  point  in  the 
plane  (11).  From  the  form 
(11)  of  the  equation  of  the 
plane  it  follows  that  there 
exist  four  immbers  jf),  ?,,  I2,  Ip 
not  all  zero,  such  that 

px,  =  l,x\^U\  +  kx"\, 
^■=l,  2,3,  4.  (13) 

In  particular,  we  have  p  ^  0, 
since  otherwise  it  would  follow 
that  (.t'),  (x-"),  and  (.^'"')  are 
'collinear  (Art.  95),  which  is 
contrary  to  hypothesis.  Con- 
versely, every  point  {x)  whose 
coordinates  are  expressible  in 
the  form  (13),  p=^0  lies  in 
the  plane  (11)  since  its  coor- 
dinates satisfy  the  equation  of 
the  plane. 

Equations  (13)  are  called  the 
parametric  equations  of  the 
plane  (11),  and  ?,,  U,  Z3  are  called 
the  homoj^eneous  parameters  of 
the  points  of  the  plane. 


u^ 

W3 

lit 

n', 

"'3 

u 

m". 

n'\ 

u 

Let  («'),  (h"),  {u"')  be  three 
given  non-collinear  planes.  The 
equation  of  the  point  deter- 
mined by  them  is  found,  by 
the  same  method  as  that  em- 
ployed in  Art.  11,  to  be 


=  0.     (12) 


Tjct  («)  be  any  plane  through 
the  point  (12).  From  the  form 
(12)  of  the  equation  of  the 
point  it  follows  that  there 
exist  four  numbers/*,  /,,  U,  I3, 
not  all  zero,  sueli  that 

i  =  l,2,3,i.  (14) 

In  particular,  we  have  2^  "^  0 
since  otherwise  it  would  follow 
that  («'),  (u"),  and  (u'")  are 
collinear  (Art.  95),  which  is 
contrary  to  hypothesis.  Con- 
versely, every  plane  (u)  whose 
coordinates  are  expressible  in 
the  form  (14),  j:)=^0  passes 
through  the  point  (12)  since  its 
coordinates  satisfy  the  equation 
of  the  point. 

Fquations  (14)  are  called  the 
parametric  equations  of  the 
point  (12),  and  l^,  L,  I3  are  called 
the  homogeneous  parameters  of 
the  planes  through  the  point. 


Arts.  94,  95]      PARAMETRIC  EQUATIONS  OP  A  LINE      115 


The*  system  of  points  (13) 
is  said  to  form  a  plane  field. 
The  equation  of  the  points  of 
this  plane  field  is  found  by  sub- 
stituting the  values  of  x^,  Xn, 
.T3,  x^  from  (13)  in  the  equation 
2«ra\  =  0  of  a  point.  The  re- 
sulting equation 


The  system  of  planes  (14)  is . 
said  to  form  a  bundle  of  planes. 
The  equation  of  tlie  planes  of 
the  bundle  is  found  by  sub- 
stituting the  values  of  Ui,  xi-i, 
«3,  ?f4  from  (14)  in  the  equation 
2«,.iv=0  of  a  plane.  The  re-' 
suiting  equation 


/,2x-',M..  +  Zo2.t'",»,  +  h^x"\u,  =  0  l^u'.x^  +  k%u'\x^  +  k'^n"\x,  =  0 

is   the    equation,    in    plane  co-  is  the  equation,  in  point  coordi- 

ordinates,    of    the    plane    field  nates,  of  the  bundle  of  planes 

(13).  (14). 

95.    Parametric   equations  of  a  line.     Range  of   points.     Pencil 
of  planes. 

Theorem.       If    (cc)    is    any  Theorem.       If    (a)    is    any 

point    on    the    line    determined  plane  through  the  line  deternmied 

by  tivo  given  c>i.stii>ct  points  (.)•')  hy  tiro  giren  distinct  planes  («') 

and  {x"),   every  determinant  of  and   {a" ),  every  determinant  of 

order  three  in  the  matrix  order  three  in  the  tnatrix 


Jby  JU-~}  iCQ  it. J 

Xi  A  it/  o  '^  3  '^ 

I  cV      1  •1/9  "^      3  *^ 

is  equal  to  zero. 

Foi',  the  points  (a;),  {x'),  {x") 
and  any  fourth  point  (x'")  are 
coplanar.  Their  coordinates 
consequently  satisfy  (11).  Since 
(11)  is  satisfied  for  all  values 
of  x"\,  x"\,  x"\,  x"\,  it  follows 
that  the  coefficient  of  each  of 
these  variables  is  equal  to  zero, 
that  is,  that  all  the  determi- 
nants'of  order  three  in  (15)  are 
equal  to  zero. 


M,               Wo 

"3 

V 

(15) 

n\      u'. 

"'3 

u' 

u'\     u'\ 

u'\ 

u 

is  eqiu 

d  to  zero. 

(16) 


For,  the  planes  (w),  {u'),  («") 
and  any  fourth  plane  ('<"')  are 
concurrent.  Their  coordinates 
consequently  satisfy  (12).  Since 
(12)  is  satisfied  for  all  values 
of  u"\,  u"\,  u"\,  u"\,  it  follows 
that  the  coefficient  of  each  of 
these  variables  is  equal  to  zero, 
that  is,  that  all  the  determi- 
nants of  order  three  in  (16)  are 
equal  to  zero. 


116 


TETRAHEDRAL   COORDINATES  [Chap.  IX. 


Conversely,  if  the  determi- 
nants of  order  three  in  (15)  are 
all  equal  to  zero,  the  points  (x), 
{x'),  and  {x")  are  collinear, 
since  they  are  coplanar  with 
any  fourth  point  (x'")  what- 
ever. 

It  follows  from  the  above 
theorem  that  there  exist  three 
numbers  j),  l^,  l^,  not  all  zero, 
such  that 


Conversely,  if  the  determi- 
nants of  order  three  in  (16)  are 
all  equal  to  zero,  the  planes  (w), 
(«'),  and  {u")  are  collinear, 
since  they  have  a  point  in  com- 
mon with  any  fourth  plane  {u'") 
whatever. 

It  follows  from  the  above 
theorem  that  there  exist  three 
numbers  p,  l^,  l^,  not  all  zero, 
such  that 


px,  =  l,x\+lix'\,  ?-=l,2,3,4.  (17)     pn^  =  l,u\+ku" .,  /=1,2,3,4.  (18) 


In  particular,  we  have  p  ^  0, 
since  otherwise  the  coordinates 
of  the  points  {x')  and  (x") 
would  be  proportional  so  that 
the  points  would  coincide. 

Equations  (17)  are  called  the 
parametric  equations  of  the  line 
determined  by  {x')  and  {x"). 
The  coefficients  l^  and  U  are 
called  the  homogeneous  param- 
eters of  the  points  on  the 
line. 

The  system  of  points  (17) 
is  said  to  form  a  range  of 
points.  The  equation  of 
the  points  of  this  range  is 
found,  by  substituting  from 
(17)  in  the  equation 

Sm.cc^  =  0 

of  a  point,  to  be 

U'^x\u-  -I-  l^'^x'^Ui  =  0. 


In  particular,  we  have  p  ^  0, 
since  otherwise  the  coordinates 
of  the  planes  (w')  and  («") 
would  be  proportional  so  that 
the  planes  would  coincide. 

Equations  (18)  are  called  the 
parametric  equations  of  the  line 
determined  by  (w')  and  {u"). 
The  coefficients  l^  and  I2  are 
called  the  homogeneous  param- 
eters of  the  planes  through 
the  line. 

The  system  of  planes  (18) 
is  said  to  form  a  pencil  of 
planes  (Art.  24).  The  equation 
of  the  planes  of  this  pencil  is 
found,  by  substituting  from 
(18)  in  the  equation 

Sw-x.  =  0 

of  a  plane,  to  be 

l{^u\Xi  +  k'Zu'^Xi  =  0. 


Arts.  95,  96]  TRANSFORMATION  117 

EXERCISES 

1.  Prove  the  following  theorems  analytically.     State  and  prove  their  duals, 
(a)   A  line  and  a  point  not  on  it  determine  a  plane. 

(6)  If  a  line  has  two  points  in  common  with  a  plane,  it  lies  in  the  plane. 

(c)  If  two  lines  have  a  point  in  common,  they  determine  a  plane. 

(d)  If  three  planes  have  two  points  in  common,  they  determine  a  line. 

2.  Write  the  parametric  equations  of  the  plane  determined  by  the  points 
(1,  7,  -  1,  3),  (2,5,  4,  1),  (10,  -  1,  -3,  -  5).  Find  the  coordinates  of  this 
plane. 

3.  Write  the  parametric  equations  of  the  point  determined  by  the  planes 
(-  5,  3,  4,  1),  (7,  -  5,  8,  2),  (6,  -  4,  —  3,  1).  Find  the  coordinates  of 
this  point. 

4.  Write  the  equation,  in  plane  coordinates,  of  the  field  of  points  in  the 
plane  xi  +  2  X2  —  Xs  —  xt  =  0. 

ScG.     First  find  the  coordinates  of  three  points  in  the  plane. 

5.  Find  the  parametric  equations  of  the  pencil  of  planes  which  pass  through 
the  two  points  Ui  —  5  W2  +  3  Ms  —  M4  =  0,  7  Mi  +  2  Mo  —  Us  —  M4  =  0. 

6.  Prove  that  the  points  (1,  2,  -  3,  -  1),  (3,  -2,  5,  -  2),  (1,  -6,  11,  0) 
are  coUinear.  Find  the  parametric  equations  of  the  line  determined  by  these 
points  and  the  equation  in  plane  coordinates  of  the  range  of  points  on  this  line. 

96.  Transformation  of  point  coordinates.  Let  (Xi,  x.,,  x^,  x^)  be 
the  coordinates  of  a  point  referred  to  a  given  system  of  tetra- 
hedral  coordinates,  so  that 

X- =aiiX  +  a^.2y  +  a.i^z  +  aj,        «  =  1,  2,  3,  4,  (19) 

in  which  the  determinant  of  the  coefficients  * 

Let  the  coordinates  of  the  same  point,  referred  to  a  second  S3^stem 
of  tetrahedral  coordinates,  be 

a;'.  =  a\yx  +  a\.jj  +  a'.^z  +  a' J,        i  =  1,  2,  3,  4,         (20) 
in  which 

A'  =  I  a'li     a'22     a'33     a'44 1  ^  0. 

*  The  sj'mbol  ]  a-^    a-^    a^    044 1  will  be  used  for  brevity  to  denote 

«ii  f'i2  ai3  "14 

the  determinant                            ^21  ^22  "23  ^'24 

Osi  ^82  Osa  034 

041  043  043  044 


118 


TETRAHEDRAL   COORDINATES  [Chap.  IX. 


It  is  required  to  determine  the  equation  connecting  the  two  sets 
of  coordinates  (x^,  x.,,  x^,  x^  and  (x\,  x'^,  x\,  x'^).  For  this  purpose 
solve  equations  (20)  for  x,  y,  z,  t.     The  results  are 

A'x  =  %A\,x\,  A'y  =  :^A\,x'„   A'z  =  2^1 ',30;'^,    A't  =  %A\,x'„ 

in  which  A'^^  is  the  cofactor  of  a'-^.  in  the  determinant  A'.  Sub- 
stitute these  values  of  x,  y,  z,  t  in  (19)  and  simplify.  The  result 
is  of  the  form 

x^  =  «ii.i-'i  +  «,„x-'.  +  «i3.»'3  +  wi^a/^, 

X2  =  (it.2iX  y  -j-  «22^  2  ~r   ^23*'^'  3  "I"  ^''24'*^  4J 
X^  =  CC^iX  y  -\-  (t^'yX  2  +  «33-'*''  3  +  '<34't'  4, 

.^4  =  a^yx\  +  a^rx".  +  aax\  +  a^^x'^, 
wherein. 

^'«,,  =  a,,A\,  +  a,2.1',2  +  «i3^1'A.3  +  nu^^'u, 

The  determinant 


T^ 


is  called  the  determinant  of  the  transformation  (21).     This  deter- 
minant is  different  from  zero,  for  if  we  substitute  in  it  the  values 
of  the  a-i^  from  (22),  we  have  at  once  *  f 
1 


(21) 


i,  k  =  l,2,  3,  4.    (22) 


«n 

«12 

«13 

«1 

«21 

«22 

«L>3 

«; 

«31 

«32 

«33 

«^ 

«41 

«42 

«43 

« 

^1'"' 


Y  4'  A' 

^i  11        ^±22        ^13 


A'    I  —  A^^  0 


*  The  product  of  two  determinants  of  order  four 

A  =  I  (111      «2-2      "33      «44  I    and  /?  =  |  ^u      622      633      644  I 

is  also  a  determiuaut  of  order  four 

C'=lcii    e.22    C33    C44I, 
in  which 

Cik  =  a.i&ii  +  ai^bki  +  CHsbkz  +  a,-46A-4,        £,  A;  =  1,  2,  3,  4. 

This  theorem  can  easily  be  verified  by  substituting  these  values  of  ak  in  C  and  ex- 
pressing V  as  the  sum  of  determinants,  every  element  of  each  being  the  product 
of  an  element  of  A  and  an  element  of  B.  Of  the  sixty-four  determinants  in  the 
sum,  forty  vanish  identically,  having  all  the  elements  of  one  column  proportional 
to  the  elements  of  another.  Each  of  the  remaining  twenty-four  determinants  has 
.B  as  a  factor.  When  the  factor  B  is  removed,  the  resulting  expression  is  the 
expansion  of  the  determinant  A. 

t  The  determinant  |  A'n  A'^^  A'^  A'^  \  whose  elements  are  the  cofactors 
of  the  elements  of  A'  is  equal  to  A'^,  as  is  seen  immediately  by  multiplying  it  by 
A'  by  the  preceding  rule,  and  simplifying  the  result. 


Arts.  96,  97]  TRANSFORMATION  119 

Since  T=f^O,  the  system  (21)  can  be  solved  for  x\,  x\,  x\,  x\  in 
terms  of  X  i,  x  2,  x  3,  x  ^.     The  results  are 


TX\  =  /3uXi  +  Al^'2  +  ^31-^-3  +  (3iiXi, 

Tx\  =  p,.x,  +  /SooXo  -I-  ^30X3  +  ^4,X4, 
Tx\  =  ^,3.7-1  -f  /3,3X-.,  4- 1833^^3  +  ^843^^4, 
Tx\  =  p,,x,  +  /3,^,  +  l3,iX,  +  /?«.T„ 


(23) 


in  which  ^^^  is  the  cofactor  of  a.^  in  the  determinant  T. 

The  transformations  (2l)  and  (23)  are  said  to  be  inverse  to  each 
other. 

97.    Transformation  of  plane  coordinates.     Let 

UiXi  +  ii^Xo  +  U3X3  +  u^Xi  =  0  (24) 

be  the  equation  of  a  given  plane,  referred  to  tlie  system  of  tetra- 
hedral  coordinates  determined  by  (19).  Let  the  equation  of  the 
same  plane,  referred  to  the  system  (20),  be 

I, \x\  4-  ti'ox',  +  u ',x',  +  H  ,x\  =  0.  (25) 

If,  in  (24),  we  replace  a;,,  x^,  x^,  x^  by  their  values  from  (21),  we 
obtain,  after  rearranging  the  terms, 

(ail?'i  +  It-zi^U  +  a^i'(3  +  «4l"4)-l''l  +(«i>"l  +  «_•_•":  +  «lj"3  +  "-i;>'A)-'>^'-2 

+  («13"l  +  «J3"2  +  «,!3"3  +  ^<43"4)-''''3  +  («14"l  +  «-:i"2  +  «34«3 

+  a^,,,).i-',=  0.  (26) 

Since  equations  (25)  and  (26)  are  the  equations  of  the  same  plane, 
their  coefficients  are  proporticmal,  hence 

l>t'\  =  «i."i  4-  (W^  +  «'."3  4-  «u"47  '■  =  1,  2,  3,  4,  (27) 

where  j^^O  is  a  factor  of  i)ruportionality.  If  we  solve  equations 
(27)  for  u^,  V.-,,  j/j,  v^,  we  have 

crn,  =  p^,n\  +  ^^,n\  +  ^.3.^-3  +  ^,,.^'4,       i  =  1,  2,  3,  4,  (28) 

in  which  o-  t^  0  and  the  p^,^  have  the  same  meaning  as  in  (23). 

Since,  when  x^,  x^,  x^,  x^  are  subjected  to  a  transformation  (21), 
M„  T<2'  "3)  "^h  ai"6  subjected  simultaneously  to  the  transformation 
(28),  the  systems  of  variables  (x)  and  (ii)  are  called  contragredient. 


120  TETRAHEDRAL   COORDINATES  [Chap.  IX. 

EXERCISES 

1.  Prove  that  the  four  planes  determined  by  equating  to  zero  the  second 
members  of  equations  (23)  are  the  faces  of  the  coordinate  tetrahedron  of  the 
system  (x'l,  x'2,  x'3,  x'4). 

2.  State  and  prove  the  dual  of  the  theorem  in  Ex.  1  for  the  second  mem- 
bers of  equations  (27). 

3.  By  means  of  equations  (21)  and  (23)  find  the  coordinates  in  each  sys- 
tem of  the  unit  point  of  the  other  system. 

4.  Determine  the  equations  of  a  transformation  of  coordinates  in  which 
the  only  change  is  that  a  different  point  is  chosen  as  unit  point. 

98.  Projective  transformations.  Equations  (21)  were  derived  as 
the  equations  connecting  tlie  coordinates  of  a  given  arbitrary  point 
referred  to  two  systems  of  tetraliedral  coordinates.  We  shall  now 
give  these  equations  another  interpretation,  entirely  distinct  from 
the  preceding  one,  but  ec^ually  important. 

Let  there  be  given  a  system  of  equations  (21)  with  determinant 
T  not  equal  to  zero.  Let  P'  be  a  given  point  and  let  its  coordi- 
nates, in  a  given  system  of  tetrahedral  coordinates,  be  {x\,  x'2,  x\, 
x'^.  By  substituting  the  coordinates  of  P'  in  the  second  members 
of  (21),  we  determine  four  numbers  a^i,  x,,  Xg,  x^,  which  we  consider 
as  the  coordinates  (in  the  same  system  of  coordinates  as  those  of 
P')  of  a  second  point  P.  To  each  point  P'  in  space  corresponds,  in 
this  way,  one  and  only  one  point  P.  Moreover,  when  the  coordi- 
nates of  P  are  given,  the  coordinates  of  P'  are  fixed  by  (23),  so 
that  to  each  point  P  corresponds  one  and  only  one  point  P'.  It  is 
useful  to  think  of  the  point  P'  as  actually  changed  into  P  by  the 
transformation  (21)  so  that,  by  means  of  (21),  the  points  of  space 
change  their  positions. 

A  transformation  determined  by  a  system  of  equations  of  the 
type  (21),  with  determinant  T  not  equal  to  zero,  is  called  a  pro- 
jective transformation.  The  projective  transformation  (23)  is 
called  the  inverse  of  (21).  If,  by  (21),  P'  is  transformed  into  P, 
then,  by  (23),  P  is  transformed  into  P'. 

By  (21),  the  points  of  the  plane  («')  are  transformed  into  the 
points  of  the  plane  (»)  determined  by  (28).  Equations  (28)  are 
called  the  equations  of  the  transformation  (21)  in  plane  coor- 
dinates. 


Arts.  98-100]  CROSS  RATIO  121 

99.  Invariant  points.  The  points  which  remain  fixed  when 
operated  on  by  a  given  projective  transformation  (21)  are  called 
the  invariant  points  of  the  transformation.  To  determine  these 
points,  put  x,  =  p.«'i  in  (21).  The  condition  on  p  in  order  that 
the  resulting  equations 

(«ii  —P)  ^\  +  «i2^"'2  +  a^zx\  +  a^iX\  =  0, 
Ojix'i  +  {0.21— P)x' 2  +  a^^x';  +  a^iX\  =  0, 
ttaiic'i  +  ttsox',  +  («33  —  1^)  ^"'3  +  a34^'4  =  0, 
a^^x'i  +  aiox\  +  a43x'3  +  {a^^  —p)  x'4  =  0 

have  a  set  of  solutions  (not  all  zero)  in  common  is  that 


D{p)  = 


an-p 

«12 

«13 

«14 

«21 

«22-i> 

«23 

«24 

"31 

«32 

«33-P 

«34 

a,. 

«42 

"43 

«44 

=  0.  (30) 


P 

Let  Pi  be  a  root  of  D{p)=  0.  If  7)1  is  substituted  forp  in  (29), 
the  points  (x')  whose  coordinates  satisfy  the  resulting  equations 
are  ifiyarrant  points  of  the  given  transformation. 

If  D{pi)  is  of  rank  three,  equations  (29)  determine  a  single 
invariant  point  when  p  =  Pi  (Art.  85).  If  D(2h)  is  of  rank  two, 
equations  (29)  detennine  a  line  when  2^  —  P\-  Each  point  of  this 
line  is  an  invariant  point  of  the  transformation.  If  D(pi)  is  of 
rank  one,  equations  (29)  determine  a  plane  of  invariant  points 
when  p=p^.  If  all  the  elements  of  D(p^)  are  zero,  every  point 
in  space  remains  fixed.  In  this  last  case,  the  transformation  is 
called  the  identical  transformation. 

100.    Cross  ratio.     The  cross  ratio  of  four  numbers  k^,  k^,  k^,  k^ 

is  defined  by  the  equation 

__  ki  —  ^2  _  ^3  —  ^2  ^ 
/Cj  —  /t^     fC^  —  K^ 

The  cross  ratio  of  four  collinear  points  Pj,  P2,  P3,  P^,  or  of  four 
collinear  planes  ttj,  ttj,  tts,  ^4,  is  equal  to  the  cross  ratio  of  the 
ratios  of  their  homogeneous  parameters  (equations  (17)  or  (18)). 
If  the  parameters  of  the  given  points  or  planes  are,  respectively, 
li,  l^'i  l\,  l\;  I" I,  V\\  r"i,  V'i,  it  follows  that  their  cross  ratio  is 
^  ^   Ul\  -  l,V,   .    V\l\  -  i\y\ 

ir\-i,v".'i"d"\-v\v\' 


122  TETRAHEDRAL   COORDINATES  [Chap.  IX. 

If  o-  =  —  1,  the  four  given  points  or  planes  are  said  to  be 
harmonic. 

An  important  property  of  the  cross  ratio  is  stated  in  the  follow- 
ing theorem : 

Theorem.  The  cross  ratio  of  four  x>oints  (or  planes)  is  equal  to 
the  cross  ratio  of  any  four  points  (or  2ilanes)  into  which  they  can  be 
projected. 

In  the  projective  transformation  (21),  let  the  points  (x')  and 
(x")  of  equation  (17)  be  projected  into  (y')  and  (y"),  respectively. 
It  follows  that  the  point  of  the  range  (17)  whose  parameters  are 
/i  and  h  is  projected  into  a  point  (?/)  of  the  range  determined  by 
(?/')  and  (y")  such  that 

yi  =  i,y'i  +  i2y"i,       ^•  =  l,  2,  3,4. 

Since  the  parameters  of  the  points  are  unchanged,  the  cross  ratio 
is  unchanged.     Similarly  for  a  set  of  four  planes  through  a  line. 

Conversely,  two  ranges  of  points,  or  pencils  of  planes,  are  pro- 
jective if  the  cross  ratio  of  any  four  elements  in  the  first  is  the 
same  as  that  of  the  corresponding  elements  in  the  second. 

^.  EXERCISES 

'  1.  Let  .4^(1,  0,  0,  0),  7^=(0,  1,  0,  0),  C=(0,  0,  1,  0),  Z)=(0,  0,  0,  1), 
E={1,  1,  1,  1).  Find  the  equations  of  a  projective  transformation  which 
interchanges  these  points  as  indicated,  determine  the  roots  of  D(  p)  —  0,  and 
find  the  configuration  of  the  invariant  elements  when 

(a)  J.  is  transformed  into  A,  B  into  i?,  C  into  C,  D  into  E^  E  into  D. 
(6)  A  is  transformed  into  B,  B  into  ^,  C  into  D,  Z>  into  C,  E  into  E. 

(c)  A  is  transformed  into  5,  B  into  C,  C  into  ^4,  Z)  into  Z>,  E  into  E. 

(d)  A  is  transformed  into  B,  B  into  C,  C  into  Z>,  B  into  £",  E  into  A. 

2.  Show  that  a  projective  transformation  can  be  found  that  will  transform 
five  given  points  A,  B,  C,  D,  E.  no  four  of  which  are  in  one  plane,  into  five 
given  points  ^4',  B',  C",  D',  E' ,  respectively,  no  four  of  which  lie  in  one 
plane.     Show  that  the  transformation  is  then  uniquely  fixed. 

^  3.  A  non-identical  projective  transformation  that  coincides  with  its  own 
inverse  is  called  an  involution.  Find  the  condition  that  the  transformation 
(21)  is  an  involution. 

^'  4.  Show  that  the  tran.sformations  X\  —  x'l,  Xo  =  x'2,  X3  =  ±  x'3,  X4  =  — x'4 
are  involutions.     Find  the  invariant  points  in  each  case. 


Art.  100]  CROSS   RATIO  123 

5.  If  P,  P'  are  any  two  distinct  corresponding  points  in  either  involution 
of  Ex.  4,  prove  the  following  statements  : 

(a)  The  line  PP'  contains  two  distinct  invariant  points  31,  31'. 
(/>)  The  points  {PP'313I')  are  harmonic. 

6.  Find  the  invariant  points  of  the  transformation  Xi  =  x'2,  x.^  =  x'3, 
3:3  =  x'4,  Xi  =  x'l.  Show  that  the  points  of  space  are  arranged  in  sets  of  four 
which  are  interchanged  among  themselves. 

'7.,  Interpret   the  equations  (Art.  36)  of    a  translation  of  axes  as  the 
etpiations  of  a  projective  transformation.     Find  the  invariant  elements. 

8.  Interpret  the  equations  (Art.  37)  of  a  rotation  of  axes  as  the  equations 
of  a  projective  transformation.  Show  how  this  transformation  can  be 
effected. 

9.  Find  the  cross  ratio  of  the  four  points  on  the  line  (17)  whose  param- 
eters are  (0,  1),  (1,  1),  (1,  5),  (4,  3). 


CHAPTER   X 

QUADRIC  SURFACES  IN  TETRAHEDRAL  COORDINATES 

101.  Form  of  equation.  Since  the  equation  F(x,  y,  z,  t)  =  0 
may  be  transformed  into  an  equation  in  tetrahedral  coordinates 
by  means  of  equation  (3)  of  Art.  88,  it  follows  that  the  equation 
of  a  quadric  surface  in  tetrahedral  coordinates  is  of  the  form 

+  2  a^iX^x^  +  2  a23.r2.r3  +  2  a^iX^Xi  +  2  Us^x^Xi  =  0.     a^^  =  a^^.  (1) 

Conversely,  any  equation  of  this  form  will  represent  a  quadric 
surface,  since  by  replacing  each  x^  by  its  value  from  (3),  Art.  88, 
the  resulting  equation  F  (x,  y,  z,  t)  =  0  is  of  the  form  discussed  in 
Chapters  VI,  VIT,  and  VIII. 

102.  Tangent  lines  and  planes.  Let  (x)  and  (y)  be  any  two 
points  in  space.  The  coordinates  of  any  point  (z)  on  the  line 
joining  (x)  to  (y)  are  of  the  form  (Art.  95) 

z^  =  \x^+l.y,,    1=1,2,3,4.  (2) 

If  (2:)  lies  on  the  quadric  A  =  0,  then 

\'A(x)  +  2Xf,A(x,y)  +  ,.'A{y)  =  0,  (3) 

wherein 
A(x,  y)  =  A(y,  x)  =  (a„?/i  +  a^,y.,  +  a3i?/3  +  a,iy,)xi  + 

(a2l2/l+  a222/2  +  «322/3  +  «422/4)^2  +  («3uVl  +  (^3^2+  ttsS^/s  +  «342/4)a'3  + 
(a4l2/l  +  a42.V2  +  0432/3  +  a442/4)^'4  =  9  2^  ^  2/.  =  ."^  A  ^^  •^'•-  ^'^^ 

If  (y)  lies  on  J.  =  0,  then  A(y)  =  0  and  one  root  of  (3)  is  X  =  0. 
If  (y)  is  so  chosen  that  both  roots  of  (3)  are  A  =  0,  we  must  have 
A(x,  y)  =  0.  If  {x)  is  regarded  as  variable,  and  A{x,  y)  is  not 
identically  zero,  the  equation  A{x,  y)  =  0  defines  a  plane.  The 
line  joining  any  point  in  this  plane  to  the  fixed  point  (y)  on  the 
quadric  A  touches  the  surface  at  the  point  (y)  (Art.  76).  The 
line  is  a  tangent  line  and  the  plane  A(x,  y)  =  0  is  a.  tangent  plane 
to  ^  =  0  at  (y). 

124 


Arts.  101-103]      INDETERMINATE   TANGENT  PLANE      125 

EXERCISES  ' 

1.  Find  the  equation  of  the  tangent  plane  to  x^  +  Xi^  +  X3-  —  a^Xi''  =  0  at 
the_point  (0,  0,  a,  1). 

2.  Show  that  equation  (4)  vanishes  identically  if 

A  =  axi^  +  bx2^  +  cxs^  =  0  and  (y)  =  (0,  0,  0,  1). 

3.  Determine  the  coordinates  of  the  points  in  which  the  line 

Xi  +  2  X2  4-  354  =  0,   ^3  —  2  X4  =  0  meets  the  surface  Xi^  —  xiX2  +  a;2X3  +  4  ^3^=  0. 

4.  Show  that  the  line  Xi  =  0,    a;i  —  3  X2  =  0  touches  the  surface 

0-4-  —  3  xr  +  bxo'^  -\-  Xi{xi  +  5  X2)  +  a;3X4  =  0. 

103.    Condition   that  the   tangent    plane   is   indeterminate.      If 

equation  (4)  is  satisfied  identically,  the  coefficient  of  each  x,  must 
vanish.     Thus  we  have  the  four  equations 

«Uyi  +  0212/2  4-  Clsilh  +  «4l2/4  =  0, 

«12yi  +  «222/2  +  032^3   +  «42y4  =  0,  /gx 

ai3^1   +   «23?/2  +  «33?/3  +   «43?/4  =  0, 

«142/l   +  «242/2  +  «342/3  +  ^uVi  =  ^^ 

If  these  equations  are  multiplied  by  yi,  y2,  Vz,  y^i  respectively,  and 
the  products  added,  the  result  is  ^(?/)=0,  hence  if  the  coordinates 
of  a  point  {y)  satisfy  all  the  equations  (o),  the  point  lies  on  the 
surface  ^  =  0.  From  (3)  it  follows  that  the  line  joining  any 
point  in  space  to  a  point  {y)  satisfying  equations  (5)  will  meet 
the  surface  ^4=0  in  two  coincident  points  at  (.?/).  If  {x)  is  any 
other  point  on  the  surface  A.  so  that  A{x)  =  0,  it  follows  from  (3) 
that  every  point  on  the  line  joining  (x)  to  (?/)  lies  on  the  surface. 
The  surface  A  is  in  this  case  singular  and  (y)  is  a  vertex  (Arts. 
66  and  67). 

Conversely,  if  A{x)  =  0  is  singular,  with  a  vertex  at  (?/),  the 
two  intersections  with  the  surface  of  the  line  joining  {y)  to  any 
point  in  space  coincide  at  {y).  The  coefficient  ^(.r,  y)  is  identi- 
cally zero  and  the  coordinates  of  {y)  satisfy  (5).  Since  these  co- 
ordinates are  not  all  zero,  it  follows  that  the  determinant 


A  = 


ttji     a, 


«13 

ai4 

(^23 

«24 

033 

n^i 

«« 

a^ 

(6) 


126  QUADRIC   SURFACES  [Chap.  X. 

vanishes.  Conversely,  if  A  =  0,  then  four  nmnbeis  y^,  j/o,  Vz,  Vi 
can  be  found  such  that  the  four  equations  (5)  ai-e  satistied.  The 
point  iy)  lies  on  A{;x)  =  0  and  in  the  plane  A{x,  y)=0.  The  line 
joining  (y)  to  any  point  {x)  will  have  two  coincident  points  in 
common  with  A(x)  =  0  at  (y)  ;  that  is,  (y)  is  a  vertex  of  the  quadric 
A.     We  thus  have  the  following  theorem : 

Theorem.  The  necessary  and  sufficient  condition  that  a  quadric 
surface  is  singidar  is  that  the  determinant  A  vanishes. 

The  determinant  A  is  called  the  discriminant  of  the  quadric  A. 
If  it  does  not  vanish,  the  quadric  will  be  called  non-singular. 
Unless  the  contrary  is  stated,  it  will  be  assumed  throughout  this 
chapter  that  the  surface  is  non-singular. 

104.  The  invariance  of  the  discriminant.  In  Chapter  VII  cer- 
tain invariants  under  motion  were  considered.  We  shall  now 
prove  the  following  theorem  which  will  include  that  of  Art.  74  as 
a  particular  case. 

Theorem  I.  If  the  equation  of  a  quadric  surface  is  subjected  to 
a  linear  transformation  (Art.  96),  the  discriminant  of  the  transformed 
equation  is  equal  to  the  product  of  the  discriminant  of  the  original 
equation  and  the  square  of  the  determinant  of  the  transformation. 

A  4 

Let  A{x)  =  2]  2  ^^ik^i^k  =  0  be  the  equation  of  a  given  quadric, 

and  let 

a;.  =  ai]X\  -f  a^^x'^  +  a^^'s  +  oLh^'a,     i  =  1,  2,  3,  4 

define  a  linear  transformation  of  non-vanishing  determinant  T. 
If  these  values  of  a;,-  are  substituted  in  A(x),  the  equation  becomes 

1=1         A=l 

in  which 

4  4 


Art.  104]  INVARIANCE   OF   DISCRIMINANT 

If  we  now  put 


127 


rik  =  X^' 


Im^^mky 


it  follows  that 


'a-=2"''^*'^- 


If  we  form  the  discriminant  A'  of  A'(x'),  we  may  write 

«ll'*ll  +  «2l'*21  +  «3l'"31  +   "4l''41  «u''l2  +  (h\'''22  +  «3l''32  +  «4l''42 

«12^11  +  «22'*21  +  «32''31  +  «42'*41  «12»'l2  +  «22^'22  +  «32'"32  +  «42''42 


A'  = 


This  determinant  may  be  expressed  as  the  product  of  two  deter- 
minants T  and  R  (Art.  96,  footnote),  thus 


«11 

«12 

«13 

«14 

^•ll 

'•l2 

''-13 

r 

"21 

«22 

«23 

«24 

»*21 

5'22 

'-23 

r 

"ai 

«32 

"33 

«34 

'31 

^-32 

^*33 

r 

«41 

«42 

«43 

«44 

ni 

*-42 

'•43 

1 

the  columns  in  the  first  factor  being  associated  with  the  columns 
in  the  second  to  form  the  elements  of  the  rows  in  the  product. 
Similarly,  the  second  factor  may  be  expressed  in  the  form 


«ll«n  +  «12«21  +  ai3f<31    +  «14«41 
«21«U  +  «22a21  +  «23«31  +  ^hi<^i\ 


«U«12  +  «12«22  +  <'l3«32  +  «14«42 
«21«12  +  «22«22  +  «23«32  +  «24«42 


which  is  the  product  of  A  by  T,  the  elements  of  the  rows  in  the 
first  factor  being  multiplied  by  the  elements  of  the  columns  in 
the  second,  hence  . ,  _  ji^k 

On  account  of  this  relation,  the  discriminant  is  said  to  be  a  rela- 
tive invariant  under  linear  transformation  of  tetrahedral  coordi- 
nates.    ISIoreover,  the  following  theorem  will  now  be  proved. 

TriKMFtEAr  IT.     Any  sth  minor  of  ^'  may  he  expressed  as  a  linear 
fiiiiciinii  nf  i]i(>_  stli  tuinors  of  ^. 


128 


QUADRIC  SURFACES 


[Chap.  X. 


The  method  of  proof  will  be  sufficiently  indicated  by  consider- 
ation of  the  minor 

This  determinant,  when  written  in  full, 


«l2'''ll  +  «22''21  +  «32'*31  +  «42'"41 


'*u''l2  "1"  ^^21^22  ~l~  '''si^Sa  ~r  C'4l'*42 
«12''l2  +  «22'*22  +  "32^32  +  a42''42 


may  be  expressed  as  the  sum  of  sixteen  determinants,  four  of 
which  vanish  identically.  The  remaining  ones  may  be  arranged 
in  pairs,  by  combining  the  determinant  formed  by  the  ith  term  of 
the  first  column  and  the  kth  term  in  the  second  with  that  formed 
by  the  kth.  term  in  the  first  column  and  the  ith  in  the  second. 
Every  such  pair  is  equivalent  to  the  product  of  a  second  minor  of 
A  and  a  second  minor  of  T.     If  /  =  2,  k=  3,  for  example,  we  have 


CC21T21        tt31^'32 
^22^21        ^32^*32 


In  this  way  it  is  seen  that  every  second  minor  of  A'  is  a  linear 
function  of  the  second  minors  of  the  determinant  R,  the  coeffi- 
cients not  containing  yv^. 

4 
By  replacing  each  ?-.^  by  its  value  ^^''w""'';  ^"^^  repeating  the 

m=l 

same  process,  it  may  be  seen  that  each  second  minor  of  R  may  be 
expressed  as  a  linear  function  of  the  second  minors  of  A,  the 
coefficients  not  containing  any  a,^..  The  same  reasoning  may  be 
applied  to  the  first  minors  of  A'.  This  proves  the  proposition. 
As  a  corollary  we  have  the  further  proposition  : 

Theorem  III.  Tlie  rank  of  the  discriminant  of  the  equation  of 
a  quadric  surface  is  riot  changed  by  any  linear  transformation  with 
non-vanishing  determinant. 


+ 

«32'' 

31        "21*'22 
31        '*22^22 

= 

«22 

«31 
«32 

'■21^32  - 

«21 
«32 

«31 
«22 

021 

«31 

»*21 

»-31 

«22 

«32 

5 

'22 

^•32 

Arts.  104,  105]       LINES  ON  THE   QUADRIC   SURFACE       129 

For,  it  follows  from  Th.  II  that  the  rank  of  A'  is  not  greater 
than  that  of  A.  Neither  can  it  be  less,  since  by  the  inverse  trans- 
formation the  minors  of  A  may  be  expressed  linearly  in  terms  of 
those  of  A'. 

We  may  now  conclude :  if  the  discriminant  A  is  of  rank  four, 
the  quadric  A(x)  =  0  is  non-singular  (Art.  103).  If  A  is  of  rank 
three,  ^  =  0  is  a  non-composite  cone,  for  if  we  take  its  vertex 
(Art.  103)  as  the  vertex  (0,  0,  0,  1)  of  the  tetrahedron  of  refer- 
ence, the  equation  A  =  0  reduces  to 

a^Xi^  +  cu.X2^  +  033X3^  +  2  a^nX^x^  4-  2  ai3cria;3  +  2  a23X2Xs  =  0. 

The  line  joining  any  point  on  the  surface  to  (0,  0,  0,  1)  lies  on 
the  surface,  which  is  therefore  a  cone  (Art.  46).  Since  by 
hypothesis  A  is  of  rank  three,  we  have 

I  «lia22«33  !  ^  ^, 

hence  the  cone  is  non-composite.  If  A  is  of  rank  two,  the  quadric 
is  composite,  for  if  we  take  two  vertices  as  (0,  0,  0,  1)  and 
(0,  0,  1,  0),  the  equation  reduces  to 

aiiX'i^  +  «22-V  +  2  ai2XiX2  =  0, 

which  is  factorable.  Since  by  hypothesis  A  is  of  rank  two, 
o„a22  —  du  is  not  zero,  hence  the  two  components  do  not  coincide. 
If  A  is  of  rank  one,  the  equation  may  be  reduced  to  the  form 
.rj2  =  0,  which  represents  a  plane  counted  twice. 

105.    Lines  on  the  quadric  surface. 

Theorem.  TTie  section  of  a  quadric  surface  made  by  any  of 
its  tangent  planes  consists  of  two  lines  passing  through  the  point 
of  tangency. 

For,  let  (?/)  be  any  point  on  a  quadric  surface  ^  =  0,  and  (z)  any 
point  on  the  tangent  plane  at  (y),  so  that  A(y)  =  0,  A(y,  z)  =  0. 
If  (2)  is  on  the  curve  of  intersection  of  A(x)  =  0,  A{x,  y)  =  0, 
then  A{z)  =  0  and  (3)  is  identically  satisfied,  hence  every  point 
of  the  line  joining  (y)  to  (2)  lies  on  the  surface.  Since  the  sec- 
tion of  a  quadric  made  by  any  plane  is  a  conic  (Art.  81)  and 
one  component  of  this  conic  is  the  line  joining  (y)  to  (2),  the 
residual  component  in  the  tangent  plane  is  also  a  straight  line. 


130  QUADRIC   SURFACES  [Chap.  X. 

The  second  line  also  passes  through  (y),  since  every  line  lying  in 
the  tangent  plane  and  passing  through  (y)  has  two  coincident 
points  of  intersection  with  the  surface  at  (y).      , 

106.   Equation  of  a  quadric  in  plane  coordinates.     Let  the  plane 

u^x^  +  UnX^  +  n^x^  +  v^Xi  =  0  (7) 

be  tangent  to  the  given  quadric  ^1,  and  let  (y)  be  its  point  of 
tangency.  Since  A(x,  y)  =  0  is  also  the  equation  of  the  tangent 
plane  at  (?/),  the  equation  2?/,;*;^  =  0  must  differ  from  A{x,  y)=0 
by  a  constant  factor  k  (Art.  24),  hence 

(hlVl  +  a2l2/2  +  «3l2/3  +  fl'41.V4  =  kUu  ^ 

«122/l  +  «222/2  +  a32.V3  +  «42i/4  =  ^'^2,  (g^) 

auVi  +  (hslh  +  «332/3  +  ttisy*  =  ^«3> 

«14^1  +  ^24^2  +  «34.V3  +  «44.V4  =  ^"4- 

Moreover,  since  (?/)  lies  in  the  tangent  plane,  we  have 

^^i2/i  +  ^'22/2  +  n^lh  +  "4?/4  =  ^-  (9) 

On  eliminating  y^,  y^,  y^,  y^  and  k  between  (8)  and  (9),  we  obtain 
as  a  necessary  condition  that  the  plane  (?<)  shall  be  tangent  to  the 
surface, 


$(?<)  = 


«11 

«21 

«31 

«12 

«22 

«32 

"13 

"23 

«33 

ai4 

«24 

«34 

u. 

U^ 

U. 

(10) 


Conversely,  if  the  coordinates  of  a  plane  (it)  satisfy  (10),  and 
if  also  A  ^  0,  then  the  plane  is  tangent  to  the  quadric  A  =  0. 
For,  if  (10)  is  satisfied,  five  numbers  y^,  y.,,  yz,  y^,  k,  not  all  zero, 
can  be  found  which  satisfy  (8)  and  (9).  In  particular,  k^O,  for 
otherwise,  since  ^=^0,  it  would  follow  from  (8)  that  y^  =  y^  = 
y^  =  y^  =  0,  contrary  to  the  hypotheses.  Since  t*,,  ?/,,  u^,  u^  are 
not  all  zero,  it  follows  from  (8)  that  y^,  y.,  y^,  y^  are  not  all  zero, 
and  hence  are  the  coordinates  of  a  point.  By  solving  (8)  for 
u^,  U2,  Us,  u^  and  substituting  in  (9),  we  obtain  ^l(//)=0,  hence 
the  point  (//)  lies  on  the  quadric  A.  From  (4)  and  (7)  it 
follows  that  the   plane  (7)    is    tangent  to  A   at    the   point    (y). 


Art.  106]  EXERCISES  131 

The  equation  ^  (»)  =  0  is  of  the  second  degree  in  ?fj,  U2,  u^,  n^. 
It  is  the  equation  of  the  quadric  in  plane  coordinates. 

By  duality  it  follows  that  any  equation  of  the  second  degree 
in  plane  coordinates,  whose  discriminant  is  not  zero,  is  the  equa- 
tion of  a  quadric  surface  in  plane  coordinates. 

If  A  is  of  rank  three,  so  that  A  =  0  is  the  equation  of  a  cone, 
the  equation  (^(a)  =0  reduces  to  C^k-u^f^  0,  2fci"i  =  0  being  the 
equation  of  the  vertex  of  the  cone.  If  A  is  of  rank  less  than 
three,  ^{u)  =  0  vanishes  identically.  The  equation  <!>(?<)  =  0  was 
in  fact  derived  simply  by  imposing  the  condition  that  the  section 
of  the  quadric  by  the  plane  (u)  should  be  composite. 

EXERCISES 

1.  If  the  equation  ^(.r)  =  0  cuntaius  but  three  variables,  show  that  it 
represents  a  singular  quadric. 

2.  Calculate  the  discriminant  of  Xi-  +  'j-{'  +  X2'^  —  x^^  =  0. 

3.  Show  that  the  di.scriuiinaiit  of  <i>(?6)  =  0  contains  the  discriminant  of 
^(x)  =  0  as  a  factor. 

4.  Given  A{x)  =  axi'-  +  bx-^^  +  cxi^  +  dx^^  =  0,  determine  the  form  of 
the  equation  $(m)  =  0. 

5.  When  the  equation  *(«)  =  0  is  given,  show  how  to  obtain  the  equation 
A{x)  =  0. 

6.  Given  A  (x)  =  axi^  +  bx-r  +  2  cxai  =  0,  find  <P(u)  =  0. 

7.  Find  the  discriminant  of 

A{X)  =  Xr  —  Xo-  —  X1X3  —  XoXs  +  XyXi  +  XoXi  +  X-iXi  =  0,. 

and  determine  the  form  of  <!>(?<)  =  0. 

8.  Given  4>(?<)  =  Ui^  —  2  uiUo  +  u-2^  +  2  ti^uz  +  2  xiiin  —  2  xi^u-i  —  2  u^u^  + 
M3-  +  »4^  +  2  u-iUi  =  0,  find  A{x)  =  0  and  interpret  geometrically.  •*  '"i"  * 

9.  Find  the  two  lines  lying  in  the  tangent  plane  Xi  =  0  to  the  quadric 
X\X-2  +  xi-  —  xi^  —  0. 

10.  \yrite  the  equation  of  a  quadric  passing  through  each  vertex  of  the 
tetrahedron  of  reference. 

11.  \Yrite  the  equation  of  a  quadric  touching  each  of  the  coordinate 
planes  (use  dual  of  method  of  Ex.  10). 

12.  Write  the  equation  of  a  quadric  which  touches  each  edge  of  the  tetra- 
hedron of  reference. 

13.  What  locus  is  represented  by  the  equation  'LatkUiUk  —  0  when  the  dis- 
criminant is  of  rank  three  ?  of  rank  two  ?  of  rank  one  ? 

14.  Show  that  through  any  line  two  planes  can  be  drawn  tangent  to  a 
given  non-singular  quadric. 


132  QUADRIC  SURFACES  [Chap.  X. 

107.  Polar  planes.  When  the  coordinates  Zj,  Z2,  z^,  Zi  of  any 
point  (z)  in  space  are  substituted  in  A(;x,  z)  =  0,  the  resulting 
equation  defines  a  plane  called  the  polar  plane  of  (z)  as  to  the 
quadric  A. 

Let  (y)  be  any  point  in  the  polar  plane  of  (2),  so  that 
A(y,  z)  =  0.     Since  the  expression 

A(y,  z)  =  A(z,  y) 

is  symmetric  in  the  two  sets  of  coordinates  yi,  y2,  y^,  2/4  and  z^,  z^, 
z^,  z^,  it  follows  that  (z)  lies  in  the  polar  plane  of  (?/).  Hence  we 
have  the  following  theorem  : 

Theorem.  If  the  j)oiut  (y)  lies  on  the  polar  plane  of  (z),  then  (2) 
lies  on  the  polar  plane  of  (y). 

Any  two  points  (?/),  (z),  each  of  which  lies  on  the  polar  plane  of 
the  other,  are  called  conjugate  points  as  to  the  quadric  A(x)  =  0. 

Dually,  any  two  planes  are  said  to  be  conjugate  if  each  passes 
through  the  pole  of  the  other. 

108.  Harmonic  property  of  conjugate  points.  We  shall  prove 
the  following  theorem. 

Theorem.  Any  two  conjugate  j^oints  (x),  (y)  and  the  two  points 
in  which  the  line  joining  them  intersects  the  quadric  constitute  a  set 
of  harmonic  jioints. 

The  coordinates  of  the  points  (z)  in  which  the  line  joining  the 
conjugate  points  (x),  (y)  as  to  the  quadric  A  are  obtained  by 
putting  Zi  =  \x-  +  /A?/-  and  substituting  these  values  in  A{z)  =  0. 
The  values  of  the  ratio  X :  fx  are  roots  of  the  equation  (Art.  102) 

A2.4(x)  +  2  XfxA(x,  y)  +  fi?A(y)  =  0. 

Since  A{x,  y)  =  0,  if  one  root  is  Aj :  /xi,  the  other  is  —  Ai :  /aj.  The 
coordinates  of  the  points  (x),  (y)  and  the  two  points  of  intersec- 
tion are  therefore  of  the  form 

^i,  Vi,   K^i  +  Mi,  K^\  -  Mi,  ?■  =  1,  2,  3,  4, 
hence,  the  four  points  ai'e  harmonic  (Art,  100). 

Dually,  any  two  conjugate  planes  («),  {v)  and  the  two  tangent 
planes  to  the  quadric  through  their  line  of  intersection  determine 
a  set  of  harmonic  planes. 


Arts.  107-110]  TANGENT   CONE  133 

109.  Locus  of  points  which  lie  on  their  own  polar  planes.     The 

condition  that  a  point  (y)  lies  on  its  own  polar  plane  A{x,  y)  =0 
as  to  A(x)  =  0  is  A{y,  y)  =  A{y)  =  0,  that  is,  that  the  point  lies  on 
the  quadric.     We  therefore  have  the  theorem  : 

Theorem.  The  locus  of  points  ichich  lie  on  their  polar  planes  as 
to  a  quadric  A(x)  =  0  is  the  quadric  itself. 

Since  when  (y)  is  a  point  on  A(x)  =  0,  A{x,  y)  =  0  is  the  equa- 
tion of  the  tangent  plane  to  A(x)  =0  at  {y),  it  follows  that  the 
polar  plane  of  any  point  on  the  surface  is  the  tangent  plane  at 
that  point. 

A  point  which  lies  on  its  own  polar  plane  will  be  said  to  be 
self-conjugate.  Dually,  a  plane  which  passes  through  its  own  pole 
will  be  said  to  be  self-conjugate. 

110.  Tangent  cone.  If  from  a  point  (y)  not  on  the  quadric  A 
all  the  tangent  lines  to  the  surface  are  drawn,  these  lines  define  a 
cone,  called  the  tangent  cone  to  ^1  from  (//). 

Theorem.  Tlie  tangent  cone  to  a  quadric  from  any  point  not  on 
the  surface  is  a  quadric  cone. 

Let  (x)  be  any  point  in  space.     The  coordinates  of  the  points 

(z)  in  which  the  line  joining  (x)  to  (y)  meets  the  quadric  A  are  of 

the  form 

2;.  =  A.i\  -|-  fxy^, 

in  which  A  :  fi  are  roots  of  the  quadric  equation 

\'A{x)  +  2  \^A(x,  y)  +  /.2.4(v/)  =  0. 

The  two  points  of  intersection  will  be  coincident  if 

[A{x,  y)J  =  A{x)A{y).  (11) 

If  now  {y)  is  fixed  and  (x)  is  any  point  on  the  surface  defined  by 
(11),  then  the  line  joining  {x)  to  {y)  will  be  tangent  to  A  =  0. 
Since  this  equation  is  of  the  second  degree  in  x,  the  theorem 
follows. 

The  curve  of  intersection  of  the  tangent  cone  from  (?/)  and  the 
quadric  is  found  by  considering  (11)  and  A{x)  =  0  simultaneous. 
The  intersection  is  evidently  defined  by 

\_A{x,y)J=0,     A{x)  =  0. 


134  QUADRIC   SURFACES  [Chap.   X. 

This  locus  is  the  conic  of  intersection  of  the  quadric  and  the  polar 
plane  of  the  point  (t/),  counted  twice. 

If  (y)  is  a  point  on  the  surface,  then  ^l(^)  =  0  and  the  tangent 
cone  reduces  to  the  tangent  plane  to  ^4  =  0  at  {y),  counted  twice. 

111.  Conjugate  lines  as  to  a  quadric.  We  shall  now  prove  the 
following  theorem. 

Theorem.  The  polar  j^layie  of  every  jwinf  of  the  line  joining  any 
two  given  points  (y),  (z)  jiasses  through  the  line  of  intersection  of  the 
polar  planes  of  {y)  and  (z). 

The  polar  planes  of  (y)  and  of  (z)  are  A{x,  y)  =  0  and  A(x,  z) 
=  0.  The  coordinates  of  any  point  of  the  line  joining  (jj)  and  (z) 
are  of  the  form  Xy--j-iJiZ-;  and  the  polar  plane  of  this  point  is 
A{x,  Xy  -\-  jxz)  =  0.  Since  this  equation  is  linear  in  Xy^  +  /x2,,  it 
may  be  rewritten  in  the  form 

XA{;x,  y)  +  fxA{x,  z)  =  0, 

which  proves  the  theorem. 

From  Art.  107  it  follows  that  the  polar  plane  of  every  point  of 
the  second  line  passes  through  the  first.  Two  such  lines  are 
called  conjugate  as  to  the  (paadric.  If  from  P,  any  point  on  the 
quadric,  the  transversal  to  any  pair  of  conjugate  lines  is  drawn,  it 
will  meet  the  quadric  again  in  the  harmonic  conjugate  of  P  as  to 
the  points  of  intersection  with  the  conjugate  lines,  since  its  inter- 
sections with  these  lines  are  conjugate  points  (Arts.  107,  108). 

EXERCISES 

1.  Determine  the  equation  of  tlie  polar  plane  of  (1,  1,  1,  1)  as  to  the 
quadric  .ii"-  +  »;2"  +  *'3^  +  ^i"  =  0. 

2.  Find  the  equation  of  the  line  conjugate  to  a:i  =  0,  a;2  =  0  as  to  the 
quadric  s'l'^  +  X2^  +  X3-  +  Xi^  =  0. 

3.  Show  that  any  four  points  on  a  line  have  tlie  same  cro.ss  ratio  as  their 
four  polar  planes. 

4.  Find  the  tangent  cone  to  3:1X2  —  x^Xt  =  0  from  the  point  (1,  2,  1,  3). 

5.  If  a  line  meets  a  ([uadric  in  P  and  Q,  show  that  the  tangent  planes  at 
P  and  Q  meet  in  the  conjugate  of  the  line. 


Arts.  111-113]  SELF-POLAR  TETRAHEDRON  135 

6.  Show  that  t]\e  quadrics  xr  +  xr  +  x-i^  —  kXi~  =  0,  xr  +  Xo-  +  x.r  —  Ix^^ 
=  0  are  such  that  the  pohxr  plane  of  (0,  0,  0,  1)  is  tlie  same  for  both.  Inter- 
pret this  fact  geometrically. 

7.  Write  the  equation  of  a  quadric  containing  the  line  Xi  =  0,  X2  =  0. 
How  many  conditions  does  this  impose  upon  the  equation  ? 

8.  Write  the  equation  of  a  quadric  containing  the  line  Xi  =  0,  x^  =  0  and 
the  line  X3  =  0,  Xi  =  0. 

9.  Show  that  through  any  three  lines,  no  two  of  which  intersect,  passes 
one  and  but  one  quadric. 

112.  Self-polar  tetrahedron.  Associated  with  every  tetrahedron 
P1P2F3P4  is  a  tetrahedron  7ri7r27r37r4  formed  by  the  polar  planes  of 
its  vertices,  ttj  of  P^,  w^,  of  P^,  ttj  of  P3,  and  -n-^  of  P^.  Conversely, 
it  follows  from  Art.  107  that  the  plane  P^P^P^  is  the  polar  plane 
of  the  point  ttittottj,  etc. 

Two  tetrahedra  P^P^P^P^,  Tr^ir^rr^-i,  such  that  the  faces  of  each 
are  the  polar  planes  of  the  vertices  of  the  other  as  to  a  given 
quadric,  are  called  polar  reciprocal  tetrahedra.  If  the  two  tetra- 
hedra coincide,  so  that  the  plane  ttj  is  identical  with  the  plane 
P^P^Pi,  etc.,  the  tetrahedron  is  called  a  self-polar  tetrahedron. 

To  determine  a  self-polar  tetrahedron  choose  any  point  Pi  not 
on  A{x)  and  determine  its  polar  plane  tt).  In  this  polar  plane 
choose  any  point  P,  not  on  A{x)  and  determine  its  polar  plane  tt^. 
This  plane  passes  through  P^  (Art.  107).  On  the  line  of  inter- 
section of  7rj7r2  choose  a  third  point  P3  not  on  A{x)  and  determine 
its  polar  plane  ^3.     The  plane  tt^  passes  through  Pi  and  P.,. 

Finally,  let  P4  be  the  point  of  intersection  of  ttittotts.  The  polar 
plane  774  of  P4  passes  through  points  P^P^P^.  Hence  the  tetra- 
hedron P^P^P^P^  =  TTi 772773774  is  a  self-polar  tetrahedron. 

113.  Equation  of  a  quadric  referred  to  a  self -polar  tetrahedron. 

Theorem.  Tlie  necessarj/  and  sufficient  condition  that  the  equa- 
tion of  a  quadric  contains  only  tlie  squares  of  the  coordinates  is  that 
a  self  polar  tetrahedron  is  chosen  as  tetrahedron  of  reference. 

If  the  tetrahedron  of  reference  is  a  self-polar  tetrahedron,  the 
polar  plane  of  the  vertex  (0,  0,  0,  1)  is  x^  =  0.  But  the  equation 
^(^>  2/)  =  <^  of  the  polar  plane  of  (0,  0,  0,  1)  is  a4ia;i  +  a42a;2  +  a43.'r3 
+  ««-'^*4  =  0,  hence  a^  =  a^  =  O43  =  0.     Since   the    polar   plane  of 


136  QUADRIC   SURFACES  [Chap.  X. 

(0,  0,  1,  0)  is  x^  =  0,  it  follows  further  that  O13  =  033  =  0,  and  since 
the  polar  plane  of  (0,  1,  0,  0)  is  X2  =  0,  that  a^z  =  0.  But  if  these 
conditions  are  all  satisfied,  then  the  polar  plane  of  (1,  0,  0,  0)  is 
Xj  =  0,  and  the  equation  of  the  quadric  has  the  form 

Conversely,  if  the  equation  of  a  quadric  has  the  form 

OtllX'j     -|-  0!22'^2      I"  '^33'^3       1     fl'44'^4    ^  "j 

the  tetrahedron  of  reference  is  a  self-polar  tetrahedron.  Since 
A^O,  the  coefficients  a^j  are  all  different  from  zero. 

If  the  coefficients  in  the  equation  of  a  quadric  are  real  numbers, 
it  follows  from  equation  (4)  that  the  polar  plane  of  a  real  point  is 
a  real  plane,  hence  from  Art.  88  the  equation  of  the  quadric  can 
be  reduced  to  the  form  2a„.'c'^  =  0  by  a  real  transformation  of 
coordinates,  that  is,  one  in  which  all  the  coefficients  in  the  equa- 
tions of  transformation  are  real  numbers. 

By  a  suitable  choice  of  a  real  unit  point  the  equation  of  the 
quadric  may  further  be  reduced  to  the  form 

Xj^  +  X2^  ±  x^  ±  x^  =  0. 

114.  Law  of  inertia.  The  equation  of  a  quadric  having  real 
coefficients  may  thus  be  reduced  by  a  real  transformation  to  one 
of  the  three  forms 

(a)  Xi^  +  xi  +  x^^  +  x^  =  0, 

(6)  xy-"  +  x,-"  +  x,""  - x-"  =  0, 

(c)  .Xi^  +  .r2-  —  x^^  —  x^^  =  0. 

Theorem.  TTie  equation  of  any  real  non-singxdar  quadric  may  he 
reducedbya  real  transformation  to  one  and  only  one  of  the  types  (a), 

(6),  (c). 

A  quadric  of  type  (a)  contains  no  real  points,  as  the  sum  of  the 
squares  of  four  real  numbers  can  be  zero  only  when  all  the  num- 
bers are  zero.  If  the  equation  is  of  type  (6),  the  surface  contains 
real  points,  but  no  real  lines,  for  a  real  line  lying  on  the  surface 
■would  cut  every  real  plane  in  a  real  point,  but  the  section  of  (b) 
by  3^4  =  0  is  the  conic  x^^  +  x^^  +  .T3'^  =  0,   which  contains  no  real 


Arts.  113-115]        RECTILINEAR  GENERATORS  137 

points.  If  the  equation  of  a  quadric  can  be  reduced  to  type  (c), 
the  surface  contains  real  points  and  real  lines.  The  line 
x^  —  X3  —  0,  X2  —  x'4  =  0,  for  example,  lies  on  the  surface.  Any 
real  plane  through  it  will  intersect  the  quadric  in  this  line  and 
another  real  line.  If  the  equation  of  a  quadric  can  be  reduced 
to  one  of  those  forms  by  a  real  transformation,  it  can  evidently 
not  be  reduced  to  either  of  the  others,  since  real  lines  and  real 
points  remain  real  lines  and  real  points. 

The  theorem  of  this  Article  is  known  as  the  law  of  inertia  of 
quadric  surfaces.  It  states  that  the  numerical  difference  between 
the  number  of  positive  terms  and  the  number  of  negative  terms 
is  a  constant  for  any  particular  equation  independently  of  what 
real  transformation  is  employed. 

By  a  transformation  which  may  involve  imaginary  coefficients 
the  equation  of  any  quadric  may  be  reduced  to  the  form  2x--^  =  0. 

For  this  purpose  it  is  necessary  only  to  replace  x-  by  — l^z  in  the 
equation  Sa^.x,^  =  0  of  Art.  113. 

115.  Rectilinear  generators.  Reguli.  If  in  the  equation 
2j;^.2  =  0,  the  transformation 

Xi  =  a;  J   -{-  X  2}      X2  =  l\X  1  —  ^2)}        ■^i^^  ^{.^  3    I"  -^  4)9        ^4  ^^  i*^  3         '''4/ 

is  made,  it  is  seen  that  the  equation  of  any  quadric  can  also  be 
written  in  the  form 

If  the  quadric  is  of  type  (c),  its  equation  can  be  reduced  to  (12) 
by  a  real  transformation.  In  the  other  cases  the  transformation 
is  imaginary. 

The  line  of  intersection  of  the  planes 

""1*^1  ~" '^2*''^3j        /CjiC^  ^^  ^2*^2  \     *^/ 

lies  on  the  quadric  for  every  value  of  k^ :  ^^,  since  the  coordinates 
of  any  point  (y)  on  (13)  are  seen  by  eliminating  k^ :  k^  to  satisfy 
(12).  Conversely,  if  the  coordinates  of  any  point  (.?/)  on  the 
quadric  are  substituted  in  (13),  a  value  of  k^ :  ko  is  determined 
such  that  the  corresponding  line  (13)  lies  on  the  quadric  and 
passes  through  [y). 


138 


QUADRIC  SURFACES 


[Chap.  X. 


No  two  lines  of  the  system  (l.'>)  intersect,  for  if  k^Xi  =  A^-jXa, 
k^x^  =  knXo,  and  k\Xi  =  ^''2^2J  k\x^  =  ^'2-^3  are  the  two  lines,  the  con- 
dition that  they  intersect  is 

k,      0      fco     0 


k\  0  k',  0 
0  k,  0  A-, 
0       k'.     0      k\ 


—  —  (kiK  2  —  "-'2"^  1 )  —  '-'• 


But  this  condition  is  not  satisfied  unless  k^  :  k,  =  A;'i :  A;',,  that  is, 
unless  the  two  lines  coincide,  hence : 

Theorem.  Througk  each  point  on  the  quadric  (12)  jjcisses  one 
and  bwt  one  line  of  the  sytitem  (13),  lying  entirehj  on  the  surface. 

A  system  of  lines  having  this  property  is  called  a  regulus 
(Art.  79). 

In  the  same  way  it  is  shown  in  the  system  of  lines 

l^x,  =  l.,x„      1^X3  =  1x2  (14) 

is  a  regulus  lying  on  the  same  quadric  (12).  Those  two  reguli  will 
be  called  the  A:-regulus  and  the  Z-regulus,  respectively.  It  was 
seen  that  no  two  lines  of  the  same  regulus  intersect.  It  will  now 
be  shown  that  every  line  of  each  regulus  intersects  every  line  of 
the  other.     Let  7.   .  _  7.  .       ^  .  _  1. 

A'j.i'j  —  A'2.^'3,        KiX^  —  /l2^2 

be  a  line  of  the  A'-regulus  and 

(j.l'l  =  1 0X4,  tj2?3  =  I2X2 

be  a  line  of  the  Z-regulus.     The  condition  that  these  lines  intersect 

^^^^^^  k,     0      A-2     0 

0      ko     0      k, 

li      0      0      l, 

0      /.      ^1      0 

But  this  equation  is  satisfied  identically  ;  hence  the  lines  intersect 
for  all  values  of  k^ :  k^  and  Zj :  l^. 

110.  Hyperbolic  coordinates.  Parametric  equations.  Each  value 
of  the  ratio  k^  :  k.,  uniquely  determines  a  line  of  the  A;-regulus  ;  each 
value  of  li  :  L  uniquely  determines  a  line  of  the  /-regulus.  These 
two  lines  intersect;  their  point  of  intersection  lies  on  the  quadric; 


=  0. 


Arts.  115-117]        PROJECTION   UPON  A  PLANE  139 

through  this  point  passes  no  other  line  of  either  regains.  Thus, 
a  pair  of  values  k^ :  A:,  and  Zj :  I2  fixes  a  point  on  the  surface. 
Conversely,  any  point  on  the  surface  fixes  the  line  of  each  system 
passing  through  it,  and  consequently  a  pair  of  values  of  Jc^ :  k^  and 
^1 :  ^2-  These  two  numbers  are  called  hyperbolic  coordinates  of  the 
point. 

From  equations  (13),  (14)  the  relations  between  the  coordinates 
x^,  X2,  X3,  x^  of  a  point  on  the  surface  and  the  hyperbolic  coordi- 
nates ki :  k.,,  li :  I2  are 

These  equations  are  called  the  parametric  equations  of  the 
quadric  (12).  Since  the  equation  of  any  non-singular  quadric  can 
be  reduced  to  the  form  (12)  by  a  suitable  choice  of  tetrahedron  of 
reference,  it  follows  that  the  general  form  of  the  parametric 
equation  of  a  quadric  surface,  referred  to  any  system  of  tetra- 
hedral  coordinates,  may  be  written  in  the  form 

ic.  =  ciijcl,  +  «-2i^\h  +  (hi^J'i  +  ^ji^'j^jj     *  =  1,  2,  3,  4. 

117.  Projection  of  a  quadric  upon  a  plane.  Given  a  quadric 
surface  A  and  a  plane  tt.  If  each  point  Pof  A  is  connected  with  a 
fixed  point  0  on  A  but  not  on  tt,  the  line  OP  will  intersect  w  in  a 
point  P',  called  the  image  of  P.  Conversely,  if  any  point  P'  in  tt 
is  given,  the  point  P  of  which  it  is  the  image  is  the  residual  point 
in  which  OP  intersects  A.  If  P  describes  a  locus  on  A,  P'  will 
describe  a  locus  on  tt,  and  conversely.  This  process  is  called  the 
projection  of  A  upon  tt. 

Through  0  pass  two  generators  ,7,  and  g^  of  A,  one  of  each 
regulus.  These  lines  intersect  tt  in  points  0^,  0,,  which  are 
singidar  elements  in  the  projection,  since  any  point  of  ^i  has  Oi 
for  its  image,  and  any  point  of  g.^  has  0^  for  its  image.  The  tan- 
gent plane  to  A  at  0  contains  the  lines  ^i,  r/o)  hence  it  inter- 
sects the  plane  tt  in  the  line  0^0,.  Any  point  P'  of  O^Oi  will 
be  the  image  of  0.     The  line  O/Jj  will  be  called  a  singular  line. 

The  tangent  lines  to  A  at  0  form  a  pencil  in  the  tangent  plane; 
any  line  of  this  pencil  is  fixed  if  its  point  of  intersection  Avith 
O1O2  is  known.  If  a  curve  on  A  passes  through  0,  the  point  in 
which  its  tangent  cuts  0^0^  will  be  said  to  be  the  image  of  the 


140  QUADRIC  SURFACES  [Chap.  X. 

point  0  on  that  curve.  The  generators  of  the  regulus  to  which  gr, 
belongs  all  intersect  g^ ;  each,  with  0,  determines  a  plane  passing 
through  g^,  and  the  intersections  of  these  planes  with  tt  is  a 
pencil  of  lines  passing  through  Oo.  Similarly  for  the  other  regu- 
lus  and  Oj.  The  two  reguli  on  A  have  for  images  the  pencils  of 
lines  in  ir  with  vertices  at  Oj,  O2. 

118.  Equations  of  the  projection.  Let  0,  Oj,  0^  be  three  vertices 
of  the  tetrahedron  of  reference ;  take  for  fourth  vertex  the  point 
of  contact  0'  of  the  other  tangent  plane  through  O1O2.     If 

0  =  (0,  0,  0,  1),  0,  =  (0,  0, 1,  0),  0'  =  (1,  0,  0,  0),  0,  =  (0,  1,  0,  0), 

the  equation  of  the  surface  may  be  written 

Let  ^1,  4)  4  be  the  coordinates  of  a  point  in  the  image  plane,  re- 
ferred to  the  triangle  of  intersection  of  o^j  =  0,  Xj  =  0,  ccj  =  0  and 
the  image  plane  tt  or  ^a^x-  =0.  Any  point  of  the  line  joining 
(0,  0,  0, 1)  to  (?/i,  2/25  Vz,  Vi)  on  A  will  have  coordinates  of  the  form 

Tcy^,  ky^,  ky^,  A'l/^  +  A, 

wherein  k'^a^y^  +  a^X  =  0  for  the  point  in  which  the  line  pierces 
the  plane  tt. 

Moreover,  since  $i  =  kyi  (i  =  1,  2,  3)  and  yiy^  —  y^^  =  0, 

,,  y^ih 7,  v?t3 

Hence,  a  point  (?/)  on  A  and  its  image  (|)  in  tt  are  connected  by 
the  equations 

Plh  =  ^l^     plh  =  Ii4.     pVz  =  44>     Plh  =  44-  (16) 

If  4  =  0,  tlien  ^1  =  0, 2/2  =  0, 2/3  =  0,  so  that  any  point  of  the  line 

0;02  corresponds  to  0.     If  ^1  =  0  and  ^  =  0,  all  the  ?/■  vanish,  but 

if  we  allow  a  point  to  approach  Oi  in  tt  along  the  line  ^  —  t^  =  0, 

then  the  corresponding  point  on  A  is 

pyx=T%\      py2  =  rt'^S      Plh  =  r^2^z,      Pl/i=i2^3, 

from  which  the  factor  ^  can  be  removed.  If  now  ^  is  made  to 
vanish,  the  point  on  A  is  defined  by 

Z/i  =  *N     2/2  =  0,     ?/3  -  T.V4  =  0. 


Arts.  117-119]       QUADRIC   THROUGH   THREE   LINES        141 

Thus,  to  the  point  Oy  correspond  all  the  points  of  the  generator 
gi,  but  in  such  manner  that  to  a  direction  |i  —  t^2  =  0  through  Oj 
corresponds  a  definite  point  (0,  0,  t,  1)  on  g^.  To  the  line 
li  —  T^2  =  0  as  a  whole  corresponds  the  line 

?/i  -  ry.  =  0,    ^3  -  ry^  =  0, 

that  is,  a  generator  of  the  regulus  g^.  A  plane  section  cut  from  A 
by  the  plane  2«iic,  =  0  has  for  image  in  tt  the  conic  whose  equa- 
tion is 

It  passes  through  Oy,  Oz- 

EXERCISES 

1.  Prove  that  if  the  image  curve  C  is  a  conic  not  passing  through  Oi  nor 
O2,  then  the  curve  C  on  J.  has  a  double 'point  at  0,  intersects  each  generator 
of  each  reguhis  in  two  points,  and  is  met  by  an  arbitrary  plane  in  four  points. 

2.  If  C  is  a  conic  through  Oi  but  not  O2,  then  C passes  through  0,  inter- 
sects each  generator  gi  in  two  points  and  each  generator  g2  in  one  point ;  it 
is  met  by  a  plane  in  three  points. 

3.  By  means  of  equations  (16),  show  that  C  of  Ex.  1  lies  on  another 
quadric  surface,  and  find  its  equation. 

4.  By  means  of  equations  (16),  show  that  C  of  Ex.  2  lies  on  another 
quadric,  having  a  line  in  common  with  A.  Find  the  equation  of  the  surface 
and  the  equations  of  the  line  common  to  both. 

119.    Quadric  determined   by  three   non-intersecting   lines.     Let 

the  equations  of  three  straight  lines  I,  I',  I",  no  two  of  which  inter- 
sect, be  respectively 

2m.x..  =  0,  ^ViXi  =  0  ;   2w>,  =  0,  ^v\x,  =  0  ;    ^u",x,  =  0,  ^v'\x,  =  0. 

It  is  required  to  find  the  locus  of  lines  intersecting  I,  I',  I". 
Let  (y)  be  a  point  on  l"  so  that 

2»",7/,.  =0,     20.  =0.  (17) 

The  equation  of  the  plane  determined  by  (_?/)  and  I  is 

2M,.?/i2v,a;.  —  2",ic.2'y,?/,  =  0,  (18) 

and  of  the  plane  determined  by  (y)  and  I'  is 

■%a\y,^v',Xi  -  •$u\xr$o',y,  =  0.  (19) 


142 


QUADRIC   SURFACES 


[Chap.  X. 


The  planes  (18)  and  (19)  intersect  in  a  line  which  intersects  I,  I', 
I".  Moreover,  the  equations  of  every  line  which  intersects  the 
given  lines  may  be  written  in  this  form.  If  we  eliminate  ?/i,  1/2, 
?/3,  2/4  from  (17),  (18),  (10),  we  obtain  a  necessary  condition  that 
a  point  (x)  lies  on  such  a  line.     The  equation  is 


u^{vx)  —  t\{ux)  U2(vx)  —  V2{vx)  U3(vx)—V3{iix)  u^(vx)~i\(vx) 

u\(v'x)  —  v\(u'x)  —  —  

u'\  u\  u'\  u'\ 

"  '""  v'\  v'' 


v 


V 


=  0, 

(20) 


wherein  (iix)  is  written  for  2m<Xj,  etc. 

Since  this  equation  is  of  the  second  degree,  the  locus  is  a  quadric. 
The  skew  lines  ?,  V,  I"  all  lie  on  it,  hence  it  cannot  be  singu- 
lar. The  common  transversals  of  I,  I',  I"  belong  to  one  regulus, 
and  I,  I',  I"  themselves  are  three  lines  of  the  other  regulus. 

If  .T,  =0,  X2  =  0  is  chosen  for  I',  and  Xj  =  0,  0:4  =  0  for  I",  the 
equation  becomes 


=  0, 


X2 

-Xi 

0 

0 

0 

0 

^i 

-X3 

Ml 

U2 

W3 

lU 

Vl 

^2 

^3 

^4 

If  we  write 


u^Vi  -  u.-y^  =  u,,i, 


this  assumes  the  form 

The  pencil  of  planes  kiX^  -f  k^x^  =  0  is  associated  with  the  pencil 
A;ia;3  +  kiX^  =  0  in  such  a  way  that  associated  planes  pass  through 
the  same  point  of  the  line  I.  Two  pencils  of  planes  associated  in 
this  way  are  projective  (Art.  100). 

The  locus  of  the  intersection  of  corresponding  planes  of  two 
projective  pencils  of  planes  whose  axes  do  not  intersect  is  a  non- 
singular  quadric  containing  the  axes  of  both  pencils. 

Dually,  the  lines  joining  the  corresponding  points  of  two  projec- 
tive ranges  generate  a  quadric  surface.  The  lines  containing  the 
given  ranges  of  points  belong  to  the  other  regulus  of  the  quadric. 
For  this  reason  it  is  sometimes  convenient  to  consider  the  gener- 
ators of  one  regulus  as  directrices  of  the  other  regulus. 


Arts.  11&-121]  THE  QUADRIC  CONE  143 

120.  Transversals  of  four  skew  lines.  Lines  in  hyperbolic  posi- 
tion. We  can  now  solve  the  problem  of  determining  the  number 
of  lines  in  space  which  intersect  four  given  skew  lines  li,  I2,  I3,  I4 
by  proving  the  following  theorem  : 

Theorem.  Four  skew  lines  have  at  least  two  (distinct  or  coin- 
cident) transversals.  If  they  have  more  than  two,  they  all  belong  to 
a  regulus. 

Any  three  of  the  lines,  as  ^1,  I2,  h,  determine  a  quadric  on  which 
^1,  I2,  I3  lie  and  belong  to  one  regains.  The  common  transversals 
of  Zj,  ?25  '3  constitute  the  generators  of  the  other  regulus.  The 
line  Z4  either  pierces  this  quadric  in  two  points  P^,  P^,  or  lies 
entirely  on  the  surface.  In  the  first  case,  through  each  of  the 
points  Pi,  P2  passes  one  generator  of  each  regulus,  hence  one  line 
meeting  /j,  h,  I3.  But  P^,  Po  are  on  l^,  hence  through  Pj,  P2  passes 
one  line  meeting  all  four  of  the  given  lines.  In  the  second  case, 
li  belongs  to  the  same  regulus  as  li,  L,  I3. 

Four  lines  which  belong  to  the  same  regulus  are  said  to  be  in 
hyperbolic  position. 

EXERCISES 

1.  Write  the  equations  of  the  quadric  deteruiined  by  the  lines 

Xi  +  X2  =  0,  X3  +  a-4  =  0  ;  2  xi  +  X2  —  X3  =  0,  X2  +  X3  —  2  X4  =  0  ; 
Xi  —  X2  —  X3  +  X4  =:  0,  xi  +  2  X2  +  3  X3  +  4  X4  =  0. 

2.  Find  the  equations  of  the  two  transversals  of  the  four  lines 

Xi  =  0,  X2  =  0  ;  X:i  =  0,  X4  =  0  ;  Xi  +  X3  =  0,  X2  +  0:4  =  0  ; 

Xi  -f-  X4  —  0,  X2  —  X-i  =  0. 

3.  When  a  tetrahedron  is  inscribed  in  a  quadric  surface,  the  tangent 
planes  at  its  vertices  meet  the  opposite  faces  in  four  lines  in  hyperbolic 
position. 

4.  State  the  dual  of  the  theorem  in  Ex.  3. 

5.  Find  tlie  polar  tetrahedron  of  the  tetrahedron  of  reference  as  to  tbe 
general  quadric  A  =  0. 

121.  The  quadric  cone.  It  has  been  seen  (Art.  104)  that  the 
surface  A  =  "^aikXiX^.  =  0  represents  a  proper  cone  if  and  only 
if  the  discriminant  is  of  rank  three.  In  this  case  there  is  one 
point  (y)  whose  coordinates  satisfy  the  four  equations 

anJh  +  «,22/2  +  «i3//3  +  oii4?/4  =  0,     i  =  1,  2,  3,  4.  (21) 

The  point  (?/)  is  the  vertex  of  the  cone. 


144  QUADRIC   SURFACES  [Chap.  X. 

The  equation  of  the  polar  plane  (Art.  107)  of  any  point  (z)  with 
regard  to  the  cone  is 

(2aa2:,)  x^  +  {^a^z,)  x^  +  C^a^^z,)  x,  +  (2a,42,)  x^  =  0.         (22) 

On  rearranging  the  equation  in  the  form 

C$a,,x,)  z,  +  (2a,,.7;,)  z,  +  {^a,,x,)  z,  +  (Sa^.x.)  z,  =  0,         (23) 

it  is  seen  that  the  coordinates  of  the  vertex  {y)  will  make  the 
coefficient  of  every  coordinate  z,-  vanish,  hence : 

Theorem  1.  TJie  j)olar  plane  of  any  point  in  space  ivith  regard 
to  a  quadric  cone  passes  through  the  vertex.  The  polar  plane  of  the 
vertex  itself  is  indeterminate. 

Moreover,  the  polar  plane  of  all  points  on  the  line  joining  any 
point  (2;)  to  the  vertex  will  coincide  with  the  polar  plane  of  (2), 
since  the  coordinates  of  any  point  on  the  line  joining  the  vertex 
(y)  to  the  point  (z)  are  of  the  form  k^y^  +  kzZi.  On  substituting 
these  values  in  (23)  and  making  use  of  (21)  we  obtain  (22)  again. 
In  particular,  if  (z)  lies  on  the  surface,  the  whole  line  (;?/)  [z)  is  on 
the  surface ;  the  polar  plane  is  now  a  tangent  plane  to  the  cone 
along  the  whole  generator  passing  through  (2:).     Hence : 

Theorem  II.  Every  tangent  plane  to  the  cone  jmsses  through  the 
vertex  and  touches  the  surface  along  a  generator. 

If  the  vertex  of  the  cone  is  chosen  as  the  vertex  (0,  0,  0,  1)  of 
the  tetrahedron  of  reference,  then  from  (22),  a^^  =  a^  =  O34  =a44=0, 
hence  the  equation  of  the  surface  is  independent  of  0:4.  Con- 
versely, if  the  equation  of  a  quadric  does  not  contain  x^,  then 
A  =  0  and  the  surface  is  a  proper  or  composite  cone  with  vertex 
at  (0,  0,  0,  1).  The  equation  of  any  quadric  cone  with  vertex  at 
(0,  0,  0,  1)  is  of  the  form 

K=  ^a^^x^x^  —  0,         i,  k=l,2,  3. 

The  equation  of  the  tangent  plane  to  K  at  a  point  (2)  is 

+  (ctsi^^i  +  a3222  +  assh)  X3  =  0. 


Arts.  121,  122]     PROJECTION  OF  A  QUADRIC   CONE        145 

If  a  plane  2"ja.\  =  0  coincides  with  this  plane,  then 
^11^1  ~r  0^1222  4"  cii^z^  =  CUi, 

^31^1  "r  <^32^2  ~r  ^33^3  ^^  ''^Sj 
^4  =  0. 

Moreover,  the  point  (z)  must  lie  in  the  plane  SWiic,  =  0,  hence 
2?<.z^  =  0.  If  z^,  Z2,  %,  ^  are  eliminated  from  these  equations,  the 
resulting  equations  are 


W4  =  0, 


an 

«12 

ai3 

u 

«21 

«22 

^23 

n 

«31 

«32 

«33 

H^ 

«1 

"2 

11-, 

0 

=  0,  (24) 


which  define  the  cone  in  plane  coordinates. 

If  the  vertex  of  the  cone  ^  =  0  is  at  the  point  (k),  then  4>(m) 
=  (2A\».)2  =  0  (Art.  106).  If  k\^0,  the  section  of  A=0  by 
the  plane  x\  =  0  is  a  conic  whose  equation  in  plane  coordinates 
is  obtained  by  equating  to  zero  the  first  minor  of  $(m)  correspond- 
ing to  a^i-  The  first  minor  of  any  element  a-^  of  the  principal 
diagonal  equated  to  zero,  together  with  ^(y()  =  0,  will,  if  l\  ^  0, 
define  the  given  cone. 

122.  Projection  of  a  quadric  cone  upon  a  plane.  Given  a  point 
0  on  a  cone  K,  but  not  at  its  vertex.  To  project  the  cone  from 
0  upon  a  plane  ir  not  passing  through  0,  connect  every  point  Pon 
^  with  0.  The  point  P'  in  which  OP  cuts  tt  is  called  tlie  ju'ojcc- 
tion  of  P  upon  tt.  Let  g  be  the  generator  of  /i" through  0,  and  0' 
the  point  in  which  g  pierces  tt.  Let  I  be  the  line  of  intersection 
of  TT  and  the  tangent  plane  along  g.  The  point  0  on  K  corre- 
sponds to  any  point  of  /,  and  to  0'  in  tt  correspond  all  the  points 
of  g.  With  these  exceptions  there  is  one-to-one  correspondence 
between  the  points  of  tt  and  of  K.  A  curve  defined  on  either 
will  uniquely  determine  a  curve  on  the  other. 

Let  /i  be  defined  by  XiX^  —  x.f=0,  v  by  x^=0,  and  O=(0,  0,  1,  0). 

If  P'  =  (ii,  $2i  ^1  ^4)?  the  coordinates  of  P  =  {Xi,  x^,  x^,  x^  are 
seen,  as  in  Art.  108,  to  be  connected  with  those  of  P'  by  the 
equations 


146  QUADRIC  SURFACES  [Chap.  X. 

EXERCISES 

1.  Show  that 

4  xi2  +  6  ri.ra  +  8  x-r  +  9  Xg^  +  12  X3X4  +4  3:42  =  0 
represents  a  cone.     Find  the  coordinates  of  its  vertex. 

2.  Find  a  value  of  k  such  that  the  equation 

x{-  —  5  x\X2  +  6  X2^  +  4  x^  —  TcXi'X'i  +  a'4"'^  =  0 
represents  a  cone. 

3.  Write  tlie  equations  of  tlie  cone  of  Ex.  1  in  plane  coordinates. 

4.  In  equations  (24),  replace  m  by  x,-  and  interpret  the  resulting  equations. 

5.  Prove  that  if  the  two  lines  of  intersection  of  a  quadric  and  a  tangent 
plane  coincide,  the  surface  is  a  cone. 

6.  What  locus  on  the  cone  K  has  for  its  projection  in  tt  a  conic  : 
(a)  not  passing  through  0'  ? 

(6)  passing  through  0',  not  touching  I  ? 
(c)  touching  I  at  O'  ? 

7.  State  some  properties  of  the  projection  upon  tt  of  a  curve  on  K  which 
passes  k  times  through  0,  has  A;'  branches  at  the  vertex,  and  intersects  g  in  n 
additional  points. 


CHAPTER   XI 
LINEAR  SYSTEMS  OF  QUADRICS 

In  this  chapter  we  shall  discuss  the  equation  of  a  quadric  sur- 
face under  the  assumption  that  the  coefficients  are  linear  functions 
of  one  or  more  parameters. 

123.  Pencil  of  quadrics.     If 

A  =  2a.,.avx-^  =  0,     B=  ^b^XiX,^  =  0 
are  the  equations  of  two  distinct  quadric  surfaces,  the  system 

.1  -  A5  =  2  (a,,  -  Xb,,)  x,x,  =  0,  (1) 

in  which  X  is  the  parameter,  is  called  a  pencil  of  quadrics. 

Every  point  which  lies  on  both  the  given  quadrics  lies  on  every 
quadric  of  the  pencil,  for  if  the  coordinates  of  a  point  satisfy  the 
equations  A  =  (),  B  =  Q,  they  also  satisfy  the  equation  A  —  XB  =  0 
for  every  value  of  X. 

Through  any  point  in  space  not  lying  on  the  intersection  of 
A  =  0,  B  =  0  passes  one  and  but  one  quadric  of  the  pencil.  If 
(y)  is  the  given  point,  its  coordinates  must  satisfy  the  equation  (1), 
hence 

A(y)-XB{y)  =  0. 

If  this  value  of  X  is  substituted  in  (1),  we  obtain  the  equation 

Aiy)B-Biy)A  =  0 

of  the  quadric  of  the  pencil  (1)  through  the  point  (y). 

124.  The  X-discriminant.  The  condition  that  a  quadric 
A  —  XB  =  0  of  the  pencil  (1)  is  singular  is  that  its  discriminant 
vanishes,  that  is, 

aji  —  A.&,i     a,2  —  A6i2 

Clfj2  —  X0i2        CI22  —  XU22 

ai3  —  AO13     a23  —  A623 

Uu  —  Xbu    a24  —  A524 

147 


I  a.i.  —  A6,.  I  = 


ai3  —  ^&i3 

«14  —  ^K 

C<23  —  AO23 

024  -  -^^24 

^33  —  •^^sa 

«34  —  ^^34 

"34  —  A63, 

Ou  -  A644 

=  0.    (2) 


148  LINEAR  SYSTEMS  OF  QUADRICS        [Chap.  XI. 

This  deterraiuant  will  be  called  the  X-discriminant.  If  it  is  iden- 
tically zero,  the  pencil  (1)  will  be  called  a  singular  pencil.  If  the 
pencil  is  not  singular,  equation  (2)  may  be  written  in  the  form 

AX*  +  4  ©A^  4-  6  4>A2  +  4  ®'A  -h  A'  =  0.  (3) 

If  A  ^  0,  this  equation  is  of  the  fourth  degree  in  A.  If  A  =  0, 
the  equation  will  still  be  considered  to  be  of  the  fourth  degree, 
with  one  or  more  infinite  roots.  It  follows  at  once  from  equation 
(3)  that  in  any  non-singular  pencil  of  quadrics  there  are  four 
distinct  or  coincident  singular  quadrics.  If  in  (3),  A  is  put  equal 
to  zero.  A'  results.  But  from  (2),  this  is  the  discriminant  of 
^  =  0.  Similarly,  A  is  the  discriminant  of  B  =  0.  Let  jS,^  be 
the  cofactor  of  6,^.  in  A.     From  (2)  and  (3)  we  obtain 

-  40  =  ciuAi  +  «22/^22  +  •••  +  (isAi- 

If  0  =  0,  yl  =  0  is  said  to  be  apolar  to  S  =  0.  Similarly,  if  0'=  0, 
B=  0  is  said  to  be  apolar  to  ^1  =  0.  A  geometric  interpretation 
of  this  property  will  be  given  later  (Art.  149). 

125.  Invariant  factors.  If  the  equations  of  the  quadrics  of  a 
non-singular  pencil  are  transformed  by  a  linear  substitution  such 
that  ^  =  0  is  transformed  into  A'  =  0  and  B  =  0  into  B'  =  0, 
then  A  —  XB  =  0  becomes  A'  —  \B'  =  0.  Moreover,  if  T  is  the 
determinant  of  the  transformation  of  coordinates,  then  (Art.  104) 

\a\,-kb',,\  =  T'\a,,-\b,,\. 
From  this  formula  we  have  at  once 

Theorem  I.  If  (A— Ai)*o  is  a  factor  of  |  a.^  —  Ai.t  |,  it  is  also  a 
factor  of  I  a',,.  —  A^'.^  |  and  conversely. 

Hence  the  numerical  value  and  multiplicity  of  every  root  of 
the  A-discriminant  is  invariant  under  any  linear  transformation 
of  coordinates.  Moreover,  by  a  proof  similar  to  that  of  Theorem 
II,  Art.  104,  we  obtain  the  following  theorem : 

Theorem  II.  Every  sth  minor  of  the  transformed  X-discriminant 
is  a  linear  function  of  the  sth  minors  of  the  original  X-discriminant 
and  conversely. 

From  the  two  theorems  I  and  II  we  obtain  at  once 


Arts.  124,  125]  INVARIANT   FACTORS  149 

Theorem  III.     If  (A  —  A,)*  is  a  factor  of  all  the  sth  minors  of 
|fl.^_X5.^|^    then    it   is   also   a  factor   of  all   the   sth   minors  of 
I  a'i*  —  -^^'i*  I  ^'^^  conversely. 
Let  (A  —  A,)*o  be  a  factor  of  the  A-discriminant, 
(X.  —  Ai/i  of  all  its  first  minors, 
(A  —  Ai)*2  of  all  its  second  minors,  etc., 
k^   being   the  highest  exponent  of   the   power  of   (A  —  Aj)    that 
divides  all  the  sth  minors,  and  k,  being  the  first  exponent  of  the 
set  that  is  zero. 
Let  also 

Li  =  A'o  —  A"i,     Lo  =  A;i  —  k^,     •••,     -i/,  =  a;,._i.  (4) 

From  Theorem  III  we  have : 
Theorem  IV.     The  expressions 

(A-AOS     (A-AOS      •••,     (A-Ai)^^ 
are  independent  of  the  choice  of  the  tetrahedron  of  reference. 

These  expressions  are  called  invariant  factors  or  elementary 
divisors  to  the  base  A  —  A,  of  the  A-discriminant. 

We  shall  next  prove  the  following  theorem  : 

Theorem  V.  The  expimcnt  of  t^ack  invariant  factor  is  at  least 
unity. 

^^^  I  f'..  -  A^,  I  =  (A  -  AO*^'i^(A), 

wliere  -F(A)  is  not  divisible  by  (A  —  Ai). 

Then  ,/ 

~  I  ",.  -  A/.,,  1  =  (A  -  X,r^-\f{\), 

where  /(A)  is  not  divisible  by  (A  —  Ai).  But  the  derivative  of 
I  tt,t  —  A/>.j.  I  with  respec-t  to  A  may  be  expressed  as  a  linear  function 
of  the  first  minors,*  and  is  conse(]uently  divisible  by  (A  — Ai)*i 
at  least. 

*  If  the  elements  of  a  determinant  \ahcd\  are  functions  of  a  variable,  it  follows 
from  the  definition  of  a  derivative  that  the  derivative  of  the  determinant  as  to  the 
variable  may  be  expressed  as  the  sum  of  determinants  of  the  form 

\u'hcd\  -\-\ah'c(l\  +  \  abc' d  \ -\- \  uhcd' \, 
in  which  a\  is  the  derivative  of  Oj,  etc. 

If  these  determinants  are  expanded  in  terms  of  the  columns  which  contain  the 
derivative,  it  follows  that  the  derivative  of  the  given  determinant  is  expressible  as 
a  linear  function  of  its  lirst  minors. 


150 


LINEAR  SYSTEMS  OF  QUADRICS        [Chap.  XI. 


Hence  7^7        1         r    \  i 

The  proof  in  the  other  cases  may  be  obtained  in  a  similar  way. 

120.  The  characteristic.  It  is  now  desirable  to  have  a  symbol 
to  indicate  the  arrangement  of  the  roots  in  a  given  A.-discriminant. 
There  may  be  one,  two,  three,  or  four  distinct  roots.  If  k^  =  1  for 
any  root  Aj,  then  L^  =  l,  and  no  other  L^  appears  for  that  factor. 
If  /Cq  =  2,  then  L^  may  be  1  or  2,  according  as  the  same  factor  is 
contained  in  all  the  first  minors  or  not.  If  all  the  exponents  L, 
associated  with  the  same  root  are  enclosed  in  parentheses  {L^, 
L2,  •••),  and  all  the  sets  for  all  the  bases  in  brackets,  the  config- 
uration is  completely  defined.  This  symbol  is  called  the  charac- 
teristic of  the  pencil  (1).     E.g.,  suppose 

and  that  X  —  X^  is  also  a  factor  of  all  the  first  minors,  but  that 
X  —  Xi  is  not.  The  characteristic  is  [2(11)].  If  A  —  Aj  is  also  a 
factor  of  all  first  minors  so  that  Lx  =  l,  Lo  =  l  to  the  base  A  —  Aj, 
the  symbol  has  the  form  [(11)(11)]. 

From  (4)  it  is  seen  that  T^i  +  Zo  -|-  •••  -{-  L^  =  A'^,  that  is,  that  the 
sum  of  the  exponents  for  any  one  root  is  equal  to  the  multiplicity 
of  that  root.  Since  the  sum  of  the  multiplicities  of  all  the  roots 
is  equal  to  four,  we  have  the  following  theorem : 

Theorem.  The  sum  of  the  exponents  in  the  characteristic  is 
always  equal  to  four. 


1.   Express  the  minor 


EXERCISES 

\h': 


of  I  a',i  —  ^ft'itl  in  terms 


I  a'23  —  X6'23       «'33 

of  the  second  minors  of  la.-jt  —  Xft,*!. 

2.    Find  the  invariant  factors  and  characteristic  of  each  of  the  following 
forms  : 


(a) 


(c) 


1-  \     0    0 

0 

0 

X 

0     0 

0        0X0 

;        i.b) 

X 

0 

1     0 

0        X     1     0 

0 

1 

0     X 

' 

0         0     0     X 

0 

0 

X     1 

0     0X0 

0 

X 

0        0 

0     0     0     X 

(d) 

X 

0 

1         0 

X     0     0     1 

' 

0 

1 

X         0 

0X10 

0 

0 

0     1 

-  \ 

Arts,  125-130]    QUADRICS  WITH   LINE   OF  VERTICES      151 

127.  Pencil  of  quadrics  having  a  common  vertex.     If  the  A-dis- 

criminant  is  identically  zero,  the  discussion  in  Arts.  124-126  does 
not  apply.  In  case  all  the  quadrics  have  a  common  vertex,  we 
may  proceed  as  follows.  If  the  common  vertex  is  taken  as 
(0,  0,  0,  1),  the  variable  Xi  will  not  appear  in  the  equation.  We 
then  form  the  A,-discriminant  of  order  three  of  the  equation  in 
a'l,  X2,  X3.  If  this  discriminant  is  not  identically  zero,  we  deter- 
mine its  invariant  factors  and  a  characteristic  such  that  the  sum 
of  the  exponents  is  three. 

Similarly,  if  the  quadrics  have  a  line  of  vertices  in  common,  we 
form  the  A.-discriininant  of  order  two,  and  a  corresponding  charac- 
teristic ;  if  the  quadrics  have  a  plane  of  vertices  in  common,  the 
A-discriminant  is  of  order  one. 

128.  Classification  of  pencils  of  quadrics.  The  principles  de- 
veloped in  the  preceding  Articles  will  now  be  employed  to  classify 
pencils  of  quadrics  and  to  reduce  their  equations  to  the  simplest 
forms.  When  the  equation  of  the  pencil  is  given,  the  charac- 
teristic is  uniquely  determined.  It  will  be  assumed  that  for  any 
given  pencil  A  —  \B  =  0,  the  A-discriminant  has  been  calculated 
and  the  form  of  its  characteristic  obtained.  For  convenience,  the 
cases  in  which  A  =  0  and  B  =  0  coincide  will  be  included  in  the 
classification,  although  in  this  case  A  —  \B  =  0  does  not  constitute 
a  pencil  as  defined  in  Art.  123. 

Since  any  two  distinct  quadrics  of  a  pencil  are  suflBcient  to 
define  the  pencil,  we  shall  always  suppose  that  the  quadric  B  =  0 
is  so  chosen  that  the  A-discriminant  has  no  infinite  roots. 

129.  Quadrics  having  a  double  plane  in  common.     By  taking  the 

plane  for  x^  =  0,  the  equation  reduces  to 

A-iX^    —  AXy    ^  U, 

A  =  \,x,\         B  =  x,\ 
and  the  characteristic  is  [1]. 

130.  Quadrics  having  a  line  of  vertices  in  common.     Let  x^  =  0, 

X2  =  0  be  the  equations  of  the  line  of  vertices.  Every  quadric 
consists  of  a  pair  of  planes  passing  through  this  line,  and  the 
equation  of  the  pencil  has  the  form 

A  —  XB  =  ajiXi^  -f  2  ai2XiX2  -f  a22X2^  —  A(6ii^i^  +  2  bi2XiX2  +  622^2^  =  0- 


152  LINEAR  SYSTEMS  OF  QUADRICS        [Chap.  XI. 

Three  cases  appear  : 

(a)  The  A-discriminant  has  two  distinct  roots  Aj,  Xj- 

(b)  The  A-discriminant  has  a  double  root  Ai,  but  not  every  first 
minor  vanishes  for  A  =  A^ 

(c)  The  A-discriminant  is  of  rank  zero  for  A  =  Aj. 

In  case  (a),  ^4  —  AjB  is  a  square  and  A  —  AjB  is  another  square. 
Let  the  tetrahedron  of  reference  be  so  chosen  that 

A-XiB  =  .r/,    ^  -  A2B  =  Xi\ 

If  we  solve  these  equations  for  ^4  and  B,  we  may,  after  a  suitable 
change  of  unit  point,  write  A,  B  in  the  form 

A  =  Aia;,^  +  X^x.^  =  0,    B  =  x''-\-  x^. 

In  case  {b)  we  have  the  relation 

(rtH&22  -  a^lKY  =  4  (aii&n  -  Ol2^'ll)(«12^22  -  Ct22&12)> 

which  is  the  condition  that  A  =  0,  5  =  0  have  a  common  factor. 
By  calling  this  common  factor  ic,,  and  the  other  factor  of  B  =  0 
(which  is  by  hypothesis  distinct  from  the  first)  2  x^,  we  may  put 

A-\^B  =  Xy^,  B  =  2x^X2. 
Solving  for  A,  B,  we  have 

In  case  (c),  we  have  ^  —  A,B  =  0,  hence  we  may  write  at  once 

Ki^'i'  +  ^2')  -  A(a;,2  +  x/)  =  0. 

The  invariant  factors  are  A  —  Aj,  A  —  Aj. 

In  this  case  we  have  then  the  following  types : 

[11  ]  A  =  Ai-Ti^  4-  x^x^^  B^xi^^  x^, 

[2]  A  =  2  AiX'iO^a  +  x^,  B  =  2  XiX2, 

[(11)J  A  =  X,{x,'  +  X,'),  B  =  X,'  +  x,\ 

131.  Quadrics  having  a  vertex  in  common.  Let  the  common 
vertex  be  taken  so  that  the  equation  of  the  pencil  contains  only 
three  variables,  .t,,  a;,,  x^.  It  will  first  be  assumed  that  the  A-dis- 
criminant is  not  identically  zero. 

Suppose  |a,.^  —  A^i,  1  =  0  has  at  least  one  simple  root  Aj.  The 
expression  A  —  XiB  is  the  product  of  two  distinct  linear  factors, 
hence  the  quadric  A  —  X^B  =  0  consists  of   two   distinct  planes, 


Arts.  130,  131]  SINGLE   C0M2>10N  VERTEX  153 

both  passing  through  the  point  (0,  0,  0,  1).  Let  the  line  of  inter- 
section of  the  planes  be  taken  for  Xj  =  0,  rcj  =  ^>  so  that  the  ex- 
pression A  —  \iB  does  not  contain  x^.     It  follows  that 

«ii  -  Kbn  =  0,     a,2  -  Ai^i2  =  0,     013  -  -^-^13  =  0- 

By  means  of  those  relations  a^,  ajj,  a^^  can  be  eliminated  from 
the  X-discriminant.     The  result  may  be  written  in  the  form 


I  Oik  -  bik  I  = 


5u(Xi  -  X)     fei2(X,  -  X)     fci3(Xi  -  X) 
bn{\i  —  A)     O22  —  X022        O23  —  X623 
bu(Ki  -  X)     0^3  -  X623        033  -  X633 


Since  Xi  was  assumed  to  be  a  simple  root  of  |  a-^  —  Xb^^.  \,  it  follows 
that  6,1  ^  0.     The  equation  of  the  pencil  now  has  the  form 

—  \{bnxi^  +  2  bi^XoXi  +  2  bi^x^Xj  +  622^2^  +  2  623a-2iC3  +  633X3^)=  0. 
If  we  make  the  substitution 

yi  =  x,  +  -i?-^- — '-^,     2/2  =  x„     y,  =  Xs, 

then  replace  t/i,  1/21  Vs  by  x^,  x^,  x^,  the  equation  of  the  pencil  takes 
the  form 

Xi^i^  +  <^(.T2,  .T3)  -  \{x,^  +f{x.^,  2:3))=  0, 

in  which  </>(x2,  x^  and  f{Xo,  2-3)  are  homogeneous  quadratic  func- 
tions of  X2,  x^.  The  above  transformation  may  be  interpreted 
geometrically  as  follows  :  Since  61,  ^t  0,  the  quadric  JB  =  0  does 
not  pass  through  the  point  (1,  0,  0,  0).     The  polar  plane 

of  the  point  (1,  0,  0,  0)  as  to  B  is  consequently  not  a  tangent 
plane  to  B  at  this  point.  The  transformation  makes  this  polar 
plane  the  new  x^,  changes  the  unit  point,  and  leaves  x^  =  0,  x^  =  0 
unchanged. 

The  expression  <^(a;2,  x^)  —  A/(a;2,  x^)  may  now  be  classified  ac- 
cording to  the  method  of  Art.  130,  and  the  associated  functions 
of  Xi,  X2,  X3  are  obtained  by  adding  X^x^^  to  (l>{x.,,  x^),  x^  to/(.T2.  Xg). 

Next  suppose  that  |a,i  — X&.^l  =0  has  no  simple  root.  It  has, 
then,  a  triple  root  which  we  shall  denote  by  Xj.  If  X— Aj  is  not 
a  factor  of  all  the  first  minors,  the  quadric  ^  — XiB  =  0  consists 


154  LINEAR  SYSTEMS   OF  QUADRICS        [Chap.  XI. 

of  two  distinct  planes.  Let  the  tetrahedron  of  reference  be 
chosen  in  such  a  way  that  these  two  planes  are  taken  as  x^  =  0, 
X3  =  0,  so  that  the  equation  of  the  quadric  has  the  form 

A  —  XiB  =  2  (a23  —  Xib23)X2X3  =  0, 
wherein  033  —  A16.23  t^  0,  but 

«ii  —  Ai&„  =  0,  Oo.,  —  XA2  =  0,  ajs  —  A1633  =  0,  ai2  —  A/>,2  =  0, 
«i3  -  ^1^13  =  0,  and 


I  dik  -  Xbifc  I   = 


6u(Xi  -  X)     6i2(X,  -  X)     b,3(X,  -  X) 

buiXl    -   X)         622(X    -    Xl)         023   -   X623 
ftlo(Xi   -  X)        023  -  Xfe23  b33(Xl   -  X) 


Since  (X  —  Ai)'  is  a  factor  of  this  determinant  and  a^^  —  Xib^s  ^  0, 
it  follows  that  ftn  =  0,  and  613^12  =  0,  that  is,  either  &13  =  0  or 
&j,,  =  0.  Since  it  is  simply  a  matter  of  notation  which  factor  is 
made  to  vanish,  let  &i3=0.  Then  612  =^0,  since  1 0,-^  —  A&j^  |  ^  0. 
Geometrically,  this  means  that  the  plane  ajj  =  0  touches  B  =  0 
along  the  line  X2  =  0,X3  =  0.  The  plane  x^  =  0  intersects  the  cone 
£  =  0  in  the  line  X2  =  0,  0^3  =  0  and  in  one  other  line.  By  a 
further  change  of  coordinates,  if  necessary,  the  tangent  plane  to 
B  =  0  along  this  second  line  may  be  taken  for  x^  =  0. 
We  then  have 

but  since 

A  =  XiB  +  2  (01,3  -  Ai&23)a^2a^3, 

we  may,  by  a  suitable  choice  of  unit  point,  write  the  equation  of 
the  pencil  in  the  form 

A-XB  =  Ai(2  xix^  +  ^-32)  +  2  XVT3  -  A(2  x^x.,  +  x^")  =  0. 

If  A  —  Ai  is  also  a  factor  of  all  the  first  minors  of  the  A-dis- 
criminant,  but  not  of  all  its  second  minors,  A  —  X^B  is  a  square 
and  represents  a  plane  counted  twice.  This  pla:ne  may  be  chosen 
for  Xj  =  0  so  that 

A  —XiB  =  (a.22  —  Xj)n-^  x.2^. 

Since  (A  —  Aj)'  is  a  factor  of   the  A-discriminant,  we  must   also 
have 

?>n^>33  -  ^'13-  =  0. 


Art.  131]  SINGLE   COMMON  VERTEX  155 

Geometrically,  this  condition  expresses  that  Xj  =  0  is  a  tangent 
plane  to  the  cone  B  =  0.     We  may  now  write 

Hence,  by  a  suitable  choice  of  unit  point,  the  equation  of  tlie 
pencil  may  be  reduced  to 

A,(2  x^,  +  x,')  +  X.?  -  A(2  x,x,  +  xi)  =  0. 

If  A  —  Ai  is  also  a  factor  of  all  the  second  minors  of  |  a^^  —  \h.^  j , 
the  equation  of  i?  =  0  is  a  multii)le  of  that  of  ^4  =  0  and  the  equa- 
tion of  the  pencil  may  be  written  in  the  form 

We  have  thus  far  supposed,  in  this  Article,  that  the  A-discrimi- 
nant  did  not  vanish  identically.  It  may  happen  that  the  deter- 
minant I  a -J.  —  A^,^. !  is  identically  zero  even  though  the  quadrics  of 
the  pencil  do  not  have  a  line  of  vertices  in  common.  In  this  case 
every  quadric  of  the  pencil  consists  of  a  pair  of  planes.  Let 
A  =  <)){Xi,  .i'o),  B=f(x.,,  X3).  Since  jai^.  — A^.-^l  is  identically  zero, 
it  follows  that 

f'll(&22^33  -  ^23')  =  0,      633(«U^'22  "  ai2')  =  0, 

and  hence  that  Ou  =  0,    ^^33  =  0,  as  otherwise  the  quadrics  would 
have  a  line  of  vertices  in  common,  contrary  to  hypothesis. 

By  an  obvious  change  of  coordinates,  we  may  write  the  equa- 
tion of  the  pencil  in  the  form  2  XjXj  —  A2  x^x^  =  0.  This  is  called 
the  singular  case  in  three  variables.  Its  characteristic  will  be 
denoted  by  the  symbol  \S\.  Collecting  all  the  preceding  results 
of  the  present  Article,  we  have  the  following  types  of  pencils  of 
quadrics  with  a  common  vertex. 

[Ill]  \,x,-  -f-  X,x,^  +  A3.T32  x-^  +  x.^  +  CC3' 

[21]  AiO-i^  +  2  Aoa-2-^3  +  x^^  x^^  +  2  XoX^ 

[1(11)]  AjXi^  +  A,(.^./  +  x,')  x,^  +  x,^  +  x,^ 

[3]  A,(2  x,x,  +  x,^)  +  2  .T,a-3  2  x,x,+x,^ 

[(21)]  A,(2  x,x,  +  x,^)  +  xi  2  x,x,  -f  x,^ 

[(111)]  X.ixC- ^  x,^  +  x,^)  .ri^  +  .r^^  +  a-3* 


156  LINEAR  SYSTEMS  OF  QUADRICS        [Chap.  XI. 

EXERCISES 

1.  Determine  the  invariant  factors  for  each  pencil  in  the  above  table. 

2.  Determine  the  nature  of  the  locus  ^  =  0,  B  =  0  for  each  pencil  in  the 
above  table. 

3.  Find  the  invariant  factors  and  the  characteristic  of  the  pencils  of 
quadric  cones  defined  by 

(a)  ^  =  3  a;r  -f  9  Xi"^  -f-  4  3:2X3  —  2  xiXs  —  6  xiXi  —  0, 
B  =  o  x{^  +  8  Xo2  -  2  xs^  —  6  XiiCs  —  14  XiXg  =  0. 

(b)  ^  =  5  Xi2  +  3  X2'^  +  2  X32  +  4  X2X3  —  2  X1X3  +  2  3:1X2  =  0, 
B  =  9  xr  -  X22  +  X32  -  4  x.:X3  +  14  X1X3  +  42  X1X2  =  0. 

(c)  A  =  5  x.^  -  5  X22  +  X32  +  6  X2X3  +  10  X1X3  -  4  X1X2  =  0, 

B  =  10  Xi2  +  2  X22  +  10  x^- 10  X2X3  +  24  X1X3  -  16  X1X2  =  0. 
{d)  A  =  2  xi2  +  2  X2^  -  2  X2X3  -  2  X1X3  =  0, 

B  =  Xi2-f  3  X.22  +  X32  —  4  X2X3  -  2  X1X3  =  0. 

4.  Find  the  form  of  the  intersection  of  vl  =  0,  jB  =  0  in  each  of  the  pencils 
of  Ex.  3. 

5.  Write  the  equations  of  each  of  the  pencils  in  Ex.  3  in  the  reduced  form. 

132.  Quadrics  having  no  vertex  in  common.  As  in  the  preced- 
ing case,  we  shall  suppose,  except  when  the  contrary  is  stated? 
that  ]  ttik  —  ^&,)fc  I  is  not  identically  zero.  If  (A  —  X{)  is  a  simple 
factor  of  the  A.-discriminant,  then  A  —  X^B  =  0  is  the  equation  of 
a  cone.  By  choosing  its  vertex  as  (1,  0,  0,  0)  and  proceeding 
exactly  as  in  Art.  131,  the  equation  may  be  reduced  to  the  form 

AiXi^  +  (fi(x2,  X3,  X4)  -  X(a:i2  +/(.C2,  0^3,  Xi))=  0. 

By  this  process  the  variable  x^  has  been  separated  and  the  func- 
tions <f>{x2,  .T3,  X4),  f(x2,  x^,  x^)  can  be  reduced  by  the  methods  of 
Art.  131,  not  including  the  singular  case. 

The  only  new  cases  that  arise  are  those  in  which  the  roots  of 
I  a,4  —  A&.jt  I  =  0  are  equal  in  pairs  or  in  which  all  four  are  equal. 

Consider  first  the  case  in  wliich  there  are  two  distinct  double 
roots  Ai  and  A25  neither  of  which  is  a  root  of  all  the  first  minors  of 
the  X-discriminant.  The  quadrics  A  —  X^B  =  0,  ^  —  X2B  =  0  are 
cones  having  distinct  vertices.  Let  the  vertex  of  the  first  be 
taken  as  (0,  0,  0,  1)  and  that  of  the  second  as  (0,  0,  1,  0).  The 
equation  of  the  former  does  not  contain  x^.     Hence,  we  have 

«i4  —  -^1^14  =  0,  a24  —  A1624  =  0,  a34  -  A1634  =  0,  a^  —  A,?>44  =  0. 


Arts.  131,  132]  NO  VERTEX  IN   COMMON  157 

When  those  values  of  a^i  are  substituted  in  |  a^^— A6,j  |  =  0,  A.  —  Aj 
is  seen  to  be  a  factor.  The  condition  that  (A  —  Ai)^  is  a  factor  is 
that  either  b^  =  0  or  that  A  —  Aj  is  a  factor  of  the  minor  cor- 
responding to  a^  —  A644.  But  in  the  latter  case  A  —  Aj  is  a  factor 
of  all  the  first  minors,  contrary  to  hypothesis,  hence  644  =  0. 
Proceeding  in  the  same  way  with  the  factor  A— A2,  it  is  seen  that 

and  also  that  633  =  0.  Hence  the  vertices  of  both  cones  lie  on  the 
quadric  B=  0.  Let  the  tangent  plane  to  5  =  0  at  (0,  0,  0,  1)  be 
taken  as  cc,  =  0,  and  the  tangent  plane  to  5  =  0  at  (0,  0,  1,  0)  be 
taken  as  x^  =  0.  Since  B  =  0  is  non-singular,  613  in  the  trans- 
formed equation  does  not  vanish,  hence  the  plane  a^j  =  0  intersects 
the  cone  A  —  \iB  =  0  in  the  line  x^  =3^2=^  and  in  another  line. 
Let  the  tangent  plane  along  this  second  line  be  taken  as  CC3  =  0; 
that  is,  make  the  transformation 

yi  =  ^i,     2/2  =  ^2, 

2  &,3(A2  -   Ai)y3  =  («!!  -    Kbn)Xl   +    2  (ai2  -  \A2)X2  +  2  &i3(A2  -  Ai).T3, 

2/4  =  ^i- 
The  equation  of  the  cone  has  now  the  form 

A-\iB=  (((02  -  Kb.„)x.^  +  2(ai3  -  kAs)^^'''.^  =  0. 

Similarly,  the  plane  .<•,  =  ()  intersects  the  cone  ^  — A2B  =  0  in 
the  line  x^  =  0,  x^  —  0  and  in  another  line.  Make  a  further  trans- 
formation by  choosing  the  tangent  plane  to  -4  —  A2B  =  0  along 
this  line  for  the  new  x^,  thus 

Vl  =  ^1,       Ih  =  ^2,       Ik  =  ^3, 
2  624(^1  —  ■^2)2/4  =(«12  —  '^2^12)-^'l  +(«22  —  A2&22)^'2  +  2  624(^1  "  •^)-'^'4- 

The  equation  of  the  second  cone  now  has  the  form 

A  —  X^B  =  ((In  —  X.bid^i^  +  2(a24  —  X^hd^^^i  =  0. 
By  a  suitable  choice  of  unit  point  the  equation  of  the  pencil  may 
be  reduced  to  . 

Ai(xi2  -f  2  X.X,)  +  A2(.r.r  +  2  x.x^)  -  Afa^i^  -f  o-J  4-2  x^x^  -\-  2  x.x^)  =  0. 

If  the  invariant  factors  are  (A  —  A,),  (A  —  Aj),  (A  —  ^y,  the  quad- 
ric A  —  Ai-B  =  0  is  a  pair  of  distinct  planes  and  as  before  A  —  A2B  =0 
is  a  cone  having  its  vertex  on  the  quadric  B  =  0.     Let  the  line  of 


158  LINEAR  SYSTEMS  OF  QUADRICS        [Chap.  XI. 

intersection  of  the  two  planes  of  ^  —  X^B  =  0  be  taken  as  x^  =  0, 
x^  =  0,  and  let  the  vertex  of  ^  —  AgB  =  0  be  at  (0,  0, 1,  0)  as  before. 
Since  this  vertex  lies  on  A  —  X^B  =  0  and  on  B  =  0,\t  lies  on  every 
quadric  of  the  pencil,  in  particular,  therefore,  on  A  —  XiB  =  0. 
Thus,  one  of  the  planes  of  the  pair  constituting  ^  —  A]B  =  0  is  the 
plane  x^  =  0.     The  other  may  be  taken  as  a^g  =  0  so  that 

A-XyB  =  (a34  -  Xib^i)XsXi  =  0. 

The  plane  x^  =  0  is  not  tangent  to  ^  —  X^B  =  0,  since  otherwise  the 
discriminant  |a,jt  —  Xb^^l  would  vanish  identically.  Hence  we  may 
choose  for  x^  =  0,  and  X2  =  0  any  pair  of  planes  conjugate  to  each 
other  and  each  conjugate  to  Xi  =  0  as  to  the  cone  A  —X2B  =  0.  The 
equation  of  the  cone  A  —  X^B  =  0  is  now 

A  —  X^B^  (a„  —  Xa&iOa^i^  +  (0^22  —  KK^^t^  +  («44  —  Kbi^x^  =  0. 

From  these  two  equations  we  may  reduce  the  equation  of  the  pen- 
cil to  the  form 

2  X^x,x,  +  X,(x,'  +  x^'-j-  X,')  -  A(2  x,x,  +  .t^^  +  x^'  +  .^•/)  =  0. 

If  (A—  A2)  is  also  a  factor  of  all  the  first  minors,  so  that  the  in- 
variant factors  are  (A  —  Ai),  (A  —  Ai),  (A  —  A2),  (A  —  A2),  the  quadrics 
A  —  Ai-B  =  0  and  A  —  A2B  =  0  both  consist  of  non-coincident  planes. 
These  four  planes  do  not  all  pass  through  a  common  point,  since 
in  that  case  all  the  quadrics  of  the  pencil  would  have  a  common 
vertex  at  that  point,  contrary  to  the  hypothesis.  We  may  conse- 
quently take 

A-X,B  =  (r<33  -  Xfi,:,)x^^  +  (a44  -  X,b^,)xi^  =  0, 
A  —  X2B  =  (ftii  —  XJ)n)Xi^  +  (a22  —  Xj32->)X2  =  0. 

By  a  suitable  choice  of  unit  point  the  equation  of  the  pencil  as- 
sumes the  form 

Ai(.r,^  +  .^-2')  +  X^x,^  +  .r/)  -  X{x,^  -f-  x.,'  +  a'a"  +  x,^)  =  0. 

The  remaining  cases  to  consider  are  those  in  which  | <x,-4  —  A6,j. | 
has  a  fourfold  factor  (A  — Ai)l  Suppose  first  that  A  —  Aj  is  not  a 
factor  of  all  the  first  minors.  The  quadric  yl  —  A,B  =  0  is  a  cone 
with  vertex  on  B~0.  Its  vertex  may  be  taken  as  (1,  0,  0,  0),  and 
the  taugent  plane  to  B  =  0  at  this  point  as  x.,  =  ^.  Since  A  — 
XiB  =  0  is  a  cone  with  vertex  at  (1,  0,  0,  0)  we  have 

aii-Xi^u  =  0,     a,2  —  A1&12  =  0..     ^'13  - '^i^a  =  0,     ai4  — Ai6i4=0. 


Art.  132] 


NO  VERTEX  IN  COMMON 


159 


Since  (1,  0,  0,  0)  lies  on  B  =  0,  we  have  b^  =  0,  and  since  the  tan- 
gent plane  at  (1,  0,  0,  0)  is  cco  =  0,  it  follows  that  613  =  0,  614  =  0. 
The  A-discriminant  now  has  the  form 


0  61  A- A) 

6i2(Ai—  X  )  CT22  —  A622 

0  «23  —  A.&23  «: 

0  -  ctiA  —  \h,      a 


0 

*^23         A.O23 
A.633 


A6, 


0 

^24  —  "■^24 
«34  -  >^hi 
a44  -  A644 


Since  (A  —  A,)*  is  a  factor  and  b^^  =^  0,  it  follows  that 


H(A-AO^  = 


I  —  A6.,. 


«34  -  A634 


The  section  of  the  pencil  of  quadrics  A  —  XB  =  0  by  the  plane 
X.J  =  0  is  the  pencil  of  composite  conies 

033X3^  -f  a^Xi^  +  2  a^^x^Xi  —  A(633a;32  4-  b^^x^^  4-  2  b^^x^x^)  =  0,     x.2  =  0. 

The  characteristic  of  this  pencil  of  composite  conies  is  [2];  it  con- 
sists (Art.  130)  of  pairs  of  lines  through  (1,  0,  0,  0)  all  of  which 
have  one  line  g  in  common.  The  plane  iC2  =  0  cuts  the  cone  A  — 
Ai-B  =  0  in  the  line  g  counted  twice,  and  g  is  defined  by  one  of  the 
factors  of  633a;3^+  2  b^^x^x^+b^^x^-,  since  it  is  common  to  all  the  conies 
of  the  pencil.  The  tangent  plane  0:2  =  0  to  J5  =  0  therefore  con- 
tains the  line  g  and  another  line  gf.  Through  the  line  g',  which 
passes  through  the  vertex  of  the  cone  A—kiB=0,  can  be  drawn 
two  tangent  planes  to  the  cone.  One  of  them  is  X2  =  0.  Choose 
the  other  for  x^  =  0.  The  plane  x^  =  0  will  touch  the  cone  A  — 
AjB  —  0  along  a  line  g".  The  plane  containing  the  two  generators 
g,  g"  of  the  cone  is  next  chosen  as  Xi  =  0.  The  equation  of  the 
cone  A  —  AjB  =  0  now  has  the  form 

A  —  AiB  =  2(a23  —  Ai623)^'2'''^3  +  («44  —  Ai644~)a;/  =  0. 

The  plane  0-3  =  0  contains  the  generator  g'  ot  B  =  0,  hence  it  is 
tangent  to  5  =  0,  and  intersects  B  =  0  in  a  line  gr,  of  the  other 
regulus.  The  plane  x^—0  contains  the  generator  ^  of  B  =  0, 
hence  meets  the  surface  in  another  line  _f/2.  The  lines  g,  g'  are  of 
opposite  systems,  hence  ^1,  g^_  belong  to  different  reguli  and  inter- 
sect. The  plane  of  g^,  g^  may  be  taken  as  the  plane  iCj  =  0.  The 
quadric  5  =  0  now  has  the  equation 

5  =  2  bxtX^x^_  -f  2  634a;3a-4  =  0. 


160  LINEAR  SYSTEMS   OF  QUADRICS        [Chap.  XI. 

By  means  of  this  equation  and  the  equation  of  the  cone  A—  \iB 
=  0  it  is  seen  that  the  equation  of  the  pencil  may  be  reduced,  by 
a  suitable  choice  of  unit  point,  to 

Xi(2  X1X2  +  2  x^x^)  +  2  x^s  +  x^^  —  A(2  x^x^  +  2  x^x^)  =  0. 

Now  suppose  A.  — Aj  is  also  a  factor  of  all  the  first  minors,  but 
not  of  all  the  second  minors.  The  surface  A  —  X^B  =  0  consists 
of  a  pair  of  planes  which  may  be  taken  for  Xs  =  0  and  ^4  =  0,  so 
that 

A-  XiB=  2(034 - Khd^i^i  =  0, 

and  A  —  \B  =  2(034  —  '^i^34)^V^4  +  (K  —  ^)  B. 

If  the  A-discriminant  is  calculated  and  the  factor  (A— Aj)^  re- 
moved, it  is  seen  that  in  order  for  |  a -^  —  Xbi^  \  to  have  the  further 
factor  (A  —  Ai)^  the  expression  611622  —  ^12^  must  vanish.  Hence 
^u^i'^  +  2  6120:1.^2  +  6223^2^  either  vanishes  identically,  or  is  a  square 
of  a  linear  expression. 

In  the  first  case,  611  =  0,  612  =  0,  6,2  =  0,  so  that  the  line  x^  =  0, 
x^  =  0  lies  on  the  quadric  B  =  0.  The  plane  ^3  =  0  passes  through 
this  line  and  intersects  B  =  0  in  a  second  line  g'.  Similarly, 
CC4  =  0  intersects  i?  =  0  in  x^  —  0  and  in  another  line  g".  Another 
tangent  plane  through  g'  may  be  taken  as  x.,  =  0,  and  the  plane  of 
g"  and  the  second  line  in  x^  =  0  as  x'l  =  0.  The  equation  oi  B  =  0 
is 

and  the  equation  of  the  pencil  may  be  reduced  to  the  form 
Ai(2  x^Xs  +  2  a-2a;4)  +  2  a;3a;4  —  A  (2  x^x^  +  2  x'2a.*4)  =  0. 

In  case  6iia;i^  +  2  6i2X'iJ*2  +  6222^2^  is  a  square,  not  identically  zero, 
the  line  x^  =  0,  .^4  =  0  touches  5  =  0  but  does  not  lie  on  it.  Let 
the  point  of  tangency  be  taken  as  (0,  1,  0,  0)  so  that  612  =  0, 
622  =  0.  If  we  now  remove  the  factor  (A  —  Ai)'  from  the  A-dis- 
criminant  and  then  put  A  equal  to  Ai,  the  result  is  603624(034— A1634). 
This  expression  is  equal  to  zero,  since  (A  —  A,)*  is  a  factor  of  the 
A-discriminant.  But  O34  — Ai634^0,  as  otherwise  A  would  be 
identical  with  B ;  hence  either  603  =  0  or  624  =  0.  Let  the  nota- 
tion be  such  that  694=  0.  Then  the  section  of  the  quadric  B  =  0 
by  the  plane  aja  =  0  consists  of  two  lines   through   (0,  1,  0,  0). 


Art.  132]  NO  VERTEX   IN   COMMON  161 

Let  L  be  the  harmonic  conjugate  of  the  line  a^j  =  0,  0:4  =  0  with 
regard  to  these  two  lines,  and  let  P  be  any  point  on  the  conic 
rr4  =  0,  ^  =  0.  If  the  plane  determined  by  P  and  L  is  chosen  for 
Xi  =  0  and  the  tangent  plane  to  B  =  0  at  P  is  taken  for  x^  =  0,  the 
equation  of  5  =  0  becomes 

B  =  biixi^  +  2  b23X2X3  +  hi^x^  =  0, 

and  the  equation  of  the  pencil  has  the  form 

Xi(.ri2  -^x^^  +  2  x,x,)  +  2  .j'3.r,  -  \  (.^i^  +  x,'  +  2  x,x,)  =  0. 

Now  suppose  that  A  —  A,  is  a  factor  of  all  the  second  minors, 
but  not  of  all  the  third  minors,  so  that  A  —  \iB  —  0  is  a  plane 
counted  twice.     Let  this  plane  be  taken  as  x^  =  0. 

vl-AiB  =  (a«-A,6«).vr  =  0. 

By  substituting  these  values  in  the  A-discriminant,  it  is  seen  that 
the  determinant  l^n'^oi^ssl  must  also  vanish  if  A  —  Aj  is  to  be 
a  fourfold  root.  This  means  tliat  the  section  of  the  quadric 
J5  =  0  by  the  plane  x^=0  consists  of  two  lines,  hence  that  cc^  =  0 
is  a  tangent  plane  to  B  =0.  Let  planes  through  these  two  lines 
be  taken  as  x^  =  0,  x.^  =  0.  The  remaining  generators  in  Xi  =  0 
and  in  ;»,  =  0  belong  to  opposite  reguli  and  therefore  intersect. 
The  plane  determined  by  them  is  now  to  be  taken  as  x^  =  0.  The 
equation  of  B  =  0  is  2  bioX^Xo  +  2  b^^x^x^  =  0,  hence  the  equation 
of  the  pencil  may  be  reduced  to  the  form 

A,  (2  x,x,  +  2  x,x,)  +  x^  -  A(2  x^x.  +  2  x^x,)  =  0. 

If  finally  A  —  Ai  is  a  factor  of  all  the  third  minors,  the  two 
equations  ^4  =  0,  B  =  0  differ  only  by  a  constant  factor.  If  B  =  0 
is  reduced  to  the  sum  of  squares  by  referring  it  to  any  self-polar 
tetrahedron,  the  equation  of  the  pencil  becomes 

\{x^  +  x,J  4-  x^  +  x^-)  —  A  x^  +  X.}  +^^32+  x{)  =  0. 

Thus  far  it  has  been  assumed  that  the  A-discriminant  did  not  iden- 
tically vanish.  Now  suppose  |  a -j  —  A6,^|  =  0  so  that  all  the  quad- 
rics  of  the  pencil  are  singular.  By  hypothesis  they  do  not  have 
a  common  vertex.  In  the  singular  pencil  two  distinct  composite 
quadrics  cannot  exist,  for,  if  ^  =  0,  B  =  0  were  composite,  we 
could  choose  A —  2  x^x^,  -B  =  2  X2,Xi,  since  the  quadrics  of  the  pencil 


162 


LINEAR  SYSTEMS  OF  QUADRICS        [Chap.  XI. 


do  not  have  a  common  vertex.  But  the  A-discriminant  of  the 
pencil  A  —  XB  —  0  is  not  identically  zero,  contrary  to  hypothesis, 
hence  the  pencil  does  not  contain  two  distinct  composite  quadrics. 
The  quadrics  ^  =  0,  B  =  0  may  therefore  be  chosen  as  cones. 
Let  the  vertex  of  ^  =  0  be  taken  as  (0,  0,  0,  1)  and  the  vertex  of 
5  =  0  as  (1,0,  0,  0). 

Let  g,  g'  be  generators  of  ^  =  0,  B  =  0  which  intersect,  but 
such  that  the  tangent  planes  along  each  of  them  does  not  pass 
through  the  vertex  of  the  other  cone.  The  plane  g,  g'  can  be 
taken  as  x^  =  0,  the  tangent  plane  to  ^  =  0  along  g  as  a-j  =  0,  and 
the  tangent  plane  to  B  =  0  along  g'  as  x^  =  0. 

The  equations  of  the  singular  quadrics  ^  =  0,  B  =  0  are  now  of 
the  form 

B  =  633.T3-  +  2  biiX^i  +  2  634X30:4  +  6440:42  =  0, 
and  the  X-discriminant  is 


I  a,-.  —  Xh,  I  = 


Since  this  expression  vanishes  identically,  the  coefficient  of  each 
power  of  X  must  be  equal  to  zero.  These  conditions  are  a^  =  0, 
644  =  0,  ai2634  —  624ai3  =  0.  The  last  condition  expresses  that  the 
planes  ai2.T2  4-  a^jX^  =  0  and  624.r2  +  634.^3  =  0  are  coincident.  By 
transforming  the  equation  of  this  plane  to  x^  =  0,  the  equation  of 
the  pencil  reduces  to 

2  XiXo  +  ax^^  —  A(2  0:20:4  +  0:3^^)  =  0. 

This  case  is  called  the  singular  case  in  four  variables.     The  char- 
acteristic will  be  denoted  by  the  symbol  [  \3\  1]. 

The  determination  of  the  invariant  factors  and  the  form  of  the 
characteristic  for  each  of  the  above  pencils  is  left  as  an  exercise 
for  the  student.  The  properties  of  the  curve  of  intersection  will 
be  developed  in  Chapter  XITI,  but  in  each  case  the  curve  is 
described  in  the  following  table  for  reference.  The  table  includes 
only  those  forms  which  do  not  have  common  double  point. 


ttu 

ai2 

«13 

0 

^12 

0 

0 

-A624 

«13 

0 

f'33  —  '^633 

-X634 

0 

-  A624 

■^34 

-A644 

Arts.  132,  133]      FORMS  OF  PENCILS  OF  QUADRICS         163 


133.     Forms  of  pencils  of  quadrics. 

Simplified  Formb  of  A  and  B 


Character- 

■     ISTIC 

[1111] 

[112] 
[11(11)] 
[13] 
[1(21)] 

[1(111)] 
[22] 


[2(11)] 


B  =  X,'  +  x.^  +  .x-a^  +  .^•4' 

A  =  Ai^'i^  +  Aa-^a^  +  2  X^x^x^  +  x^ 
B  =  x^^  +  x.y^  +  2  x^Xi 

A  =  Aia',2  +  A^a-.^^  +  X.ix-'  +  x,^) 
B  =  X,' +  x,^  +  x-' +  x,^ 

A  =  AiXi^  H-  A2(2  x-o.-Tj  +  x,^)  +  2  x,x, 

^  =  AiXi^  +  A2(2  x,x,  +  a-,2)  +  x,"^ 
B  =  x,^  +  2  oj^ajg  +  .r/ 


A  =  AjOJi^  +  A^CiCo^  +  X32  +  X42) 
£  =  x^"^  +  a-/  +  x^^  +  a;^^ 


A2  (.1-/  +  2  aJiXa) 


A  =  Ai(a-i2  +  a;,^  +  x^^)  +  2  AoiKga'^ 


[(11)(11)]  A  =K{x,'  +  x.^)  +  \,(x,^  +  X,') 
B=x,'  +  x,'  +  x-'  +  x,- 


Curve  op  Intersection 

OF   ^=  0   AND    ^=0 

A  general  space 
quartic  of  the 
first  species. 

A  nodal  quai'- 
tic. 

Two  conies 
which  intersect  at 
two  distinct 
points. 

A  cuspidal 
quartic. 

Two  conies 
which  touch  each 
other. 

A  conic  counted 
twice.  At  each 
point  of  this  conic 
the  quadrics  are 
tangent. 

A  generator  and 
a  space  cubic.  The 
generator  and  the 
cubic  intersect  in 
distinct  points. 

Two  intersect- 
ing generators, 
and  a  conic  which 
intersects  each 
generator.  The 
three  points  of  in- 
tersection are  dis- 
tinct. 

Four  generators 
which  intersect  at 
four  points. 


164 


LINEAR  SYSTEMS  OF  QUADRICS        [Chap.  XI. 


CllARAO- 
TEKISTIC 


[^] 


Simplified  Fokms  of 

A   AND  B 


A  =  Xi{2  X1X2  +  2  x^Xi) 
-f  2  x^,  +  x,^ 


Curve  of  Intf.rsection  op 

^  =  0   AND   ^=0 

A  generator  and  a  space 
cubic.  The  generator 
touches  the  cubic. 


[(22)] 


[(31)] 


A 
B. 


A 
B 


[(211)]  A: 

B. 

[(1111)]     A 
B 


[13|1]        A 
B 


Xi(2  XyX^  -\-  2  x^Xi)  +  2  x^x^ 


"T  -^  ^3'*'4 


Ai(^  ^'1^2  ~f"  -^  X^X^j  -\-  Xi 

x,^  +  x,^  + X,' +  x,^ 


2  .rox^  +  .Ts^ 


Three  generators,  one 
counted  twice.  This 
generator  intersects 
each  of  the  others. 

Two  intersecting  gener- 
ators and  a  conic 
which  touches  the 
plane  of  the  generators 
at  their  point  of  inter- 
section. 

Two  intersecting  gener- 
ators each  counted 
twice.  The  quadrics 
touch  at  each  point  of 
each  generator. 

The  quadrics  coincide. 

A  conic  and  a  generator 
counted  twice.  The 
vertices  of  the  cones 
all  lie  on  this  gen- 
erator. 


EXERCISES 

1.  Derive  the  invariant  factors  of  each  of  the  above  systems  of  quadrics. 

2.  Find  the  equations  of  each  conic  and  eacli  rectilinear  generator  of  in- 
tersection of  the  quadrics  of  the  above  pencils. 

3.  Determine  the  invariant  factors;  find  the  equations  of  the  curve  of 
intersection,  and  write  the  equations  in  the  reduced  form  of  the  pencils 
determined  by 

A  -  Xi2  —  a-u2  4.  2  X32  +  2  x^  +  5  XiX.^  =  0, 

L'  =  3  a*]-  —  x-r  +  xi^  —  8  xi^  —  2  xiXo  —  2  x^Xi  =  0. 


(a) 


AuTs.  133,  134]        LINE   CONJUGATE  TO  A  POINT  165 

.^s       A  =  x{^  +  Xi^  +  4  3:32  +  X42  +  4  xiX-2  +  <3  X2X3  +  4  XiX^  =  0, 

B  =  X2'^  +  SX3^  +  Xi^  +  2XiX3  +  2x.iX3^0. 

,.       A  =  3  xr  -  X2^  -  2  X32  +  2  X42  +  2  xiXg  —  4!ciX3  =  0, 

B  =  i  xr'  -  X22  +  2  X32  +  3  X42  +  2  XiX2  +  2  xix^  +  4  X3X4  =  0. 

/^N       ^  =  3  Xr  +  2  X22  —  Xs'^  —  X42  +  4  X1X2  —  2  X3X4  =  0, 

i>  =  3  Xi'-'  —  ^2^  —  X32  —  X42  +  X1X2  —  2  X3X4  —  3  X2X4  —  3  X1X4  =  0. 

4.    To  what  type  does  a  pencil  of  concentric  spheres  belong  ?     A  pencil  of 
tangent  spheres  ? 

134.    Line   conjugate  to  a  point.     The    equation  of   the   polar 
plane  of  a  point  (y)  with  respect  to  any  quadric  of  the  pencil  (1)  is 

As  A  varies,  this  system  defines  a  pencil  of  planes  (Art.  24).  The 
axis  of  the  pencil,  namely  the  line 

is  said  to  be  conjugate  to  the  point  (y)  as  to  the  pencil  of  quadrics. 
Let  (y)  describe  a  line,  two  points  of  which  are  (/)  and  (y"). 
It  is  required  to  find  the  locus  of  the  conjugate  line.     Since 

y.  =  H-iy'i  +  M"i,     i  =  1,  ^,  3,  4 

(Art.  95),  the  line  conjugate  to  (?^'is,  by  definition, 

H-i'^aiky'i^k  +  H-2^aiky"iXk  =  0,  '  ixyV),kU\^k  +  H"i^b,,y'\x,  =  0. 

As  (y)  describes  the  line  joining  {y')  to  (y")  the  ratio  fx^ :  ^uo  takes 
all  possible  values.  If  between  these  equations  (x.^ :  fi.,  is  elimi- 
nated, the  resulting  equation  defines  the  quadric  surface 

%a,^\x, .  26,,y",a;,  -  %a,,y",x,  ■  ^j,^\x,  =  0.  (5) 

From  the  method  of  development  it  follows  (Art.  119)  that  all  the 
lines  of  the  system  belong  to  one  regulus  (Art.  115). 

The  polar  planes,  with  respect  to  a  given  quadric  of  the  pencil, 
of  two  fixed  points  {y'),  (y")  on  the  given  line  intersect  in  the  line 

2a.y.a;,  -  \%h,^\x,  =  0,     2a,^",ar,  -  X^b,,y",x,  =  0. 

If  between  these  equations  A  is  eliminated,  the  resulting  equation 
defines  the  same  quadric  (5).  From  Art.  115  it  follows  that  this 
second  system  of  lines  constitutes  the  other  regulus  on  the  surface. 


(Ill  —  A^u 

((12  —  A612 

«13  -  A&13 

«14-A6i4 

Wl 

<^i2  —  •^^^la 

({22  —  A622 

(123  -  A623 

Cl'24  "~~  '^^24 

M2 

«13  -  'V_>i3 

(123  —  AO23 

'^^33  —  AO33 

«34  -  A634 

^*3 

«14  —  -^^14 

O24  —  A&24 

(^34  —  Atl34 

044  -  Af'44 

«4 

Ml 

%U 

U, 

Ui 

0 

IGG  LINEAR  SYSTEMS   OP  QUADRICS        [Chap.  XI. 

135.  Equation  of  the  pencil  in  plane  coordinates.  Let  A  —XB=  0 
be  the  erjuation  of  a  iion-singular  pencil  of  quadrics.  The 
equation 


=  0         (6) 


expresses  the  condition  that  the  section  of  a  quadric  of  the  pencil 
by  a  plane  (w)  is  composite  (Art.  106).  For  a  given  value  Aj  of  A, 
(6)  is  the  equation  of  the  quadric  ^  — Aii5  =  0  in  plane  coordi- 
nates, if  it  is  non-singular.  If  ^  — AiJ3  =  0  is  a  cone,  (6)  is  the 
equation  of  its  vertex  counted  twice.  If  A  —  AjB  =  0  is  composite, 
(6)  vanishes  identically. 

Equation  (6)  is  called  the  equation  of  the  pencil  in  plane  coor- 
dinates.    Arranged  in  powers  of  A,  it  is  of  the  form 

$i(m)A'  +  3  ^,{u)X^  -t-  3  *2(")^  +  ^2('0  =  0.  (7) 

If  $i(rf)  ^  0,  the  equation  is  of  the  third  degree  in  A.  When  (7) 
is  not  identically  zero,  it  will  be  said  to  be  a  cubic  in  any  case, 
even  if  it  has  one  or  more  infinite  roots.  Hence  we  have  the  fol- 
lowing theorem : 

Theorem.  Every  plane  intersects  three  distinct  or  coincident 
quadrics  of  a  non-singtdar  j}encil  in  composite  conies. 

The  coefficient  of  each  power  of  A  in  (7)  is  homogeneous  and  of 
the  second  degree  in  Wi,  U2,  n^,  v^  (if  it  is  not  identically  zero),  hence, 
when  equated  to  zero,  it  defines  a  quadric  in  plane  coordinates. 
Since  the  pencil  is  non-singular,  we  may,  without  loss  of  general- 
ity, assume  that  the  quadrics  ^=0,  andi?  =  0  are  non-singular 
(Art.  128).  The  equation  ^.^(n)  =  0  is  seen,  by  putting  A  =  0  in 
(6),  to  be  the  equation  of  ^  =  0  in  plane  coordinates.  An  analo- 
gous statement  holds  for  <I>i(?/)  =  0  and  B  =  0.  The  geometric 
meaning  of  the  other  coefficients  will  be  discussed  later  (Art. 
149). 


Arts.  135,  136]  BUNDLE   OF  QUADRICS  167 

EXERCISES 

1.  Write  the  equation  in  plane  coordinates  of  the  pencil  of  quadrics 
x{^  -  xr  +  :c3^  +  oXi^  —  6  xiXi  +  i  xsXi  —  X(2  X2X4  4  xr  +  xo^  +  xs^)  =  0. 

2.  Determine  the  equations  of  the  three  quadrics  of  the  pencil  of  Ex.  1 
which  touch  the  plane  3:4  =  0. 

3.  Determine  equation  (7)  for  the  pencil 

a(2  X1X2  +  2  a;3X4)  +  x{^  —  X(2  Xix.y  +  2  X3X4)  =  0. 

Show  that  (7)  vanishes  identically  for  each  of  the  planes  Xi=  0,  X3  =  0, 
X4  =  0,  and  interpret  the  fact  geometrically. 

136.  Bundle  of  quadrics.  If  ^1  =  '^a^i^x^x^  =  0,  B  =  Vj^^x^x^  =  0, 
C  =  %Ci^x-x^  =  ()  are  three  given  quadrics  which  do  not  belong  to 
the  same  pencil,  the  system  defined  by  the  equation 

Ai^  +  A,B  +  A3  6'  =  0,  (8) 

in  which  Xy,  A,?  ^3  are  parameters,  is  called  a  bundle  of  quadrics. 
The  three  given  quadrics  ^  =  0,  J3  =  0,  C=0  intersect  in  at  least 
eight  distinct  or  coincident  points,*  through  each  of  which  pass 
all  the  quadrics  of  the  bundle.  These  eight  points  cannot  be 
taken  at  random,  for  in  order  that  a  quadric  shall  pass  through 
eight  given  points,  the  coordinates  of  each  point  must  satisfy  its 
equation,  thus  giving  rise  to  eight  linear  homogeneous  equations 
among  the  coefficients  in  the  equation  of  the  quadric.  If  the 
eight  given  points  are  chosen  arbitrarily,  these  eight  equations  are 
independent  and  the  system  of  quadrics  determined  by  them  is  a 
pencil. 

It  is  seen  that  seven  given  arbitrarily  chosen  points  determine 
a  bundle  of  quadrics  passing  through  them.  Since  all  the  quadrics 
of  the  bundle  have  at  least  one  fixed  eighth  point  in  common,  we 
have  the  following  theorem  : 

Theorem  I.  All  the  quadric  surjaces  ivhich  pass  through  seven 
independent  2^0 i) its  in  space  pass  through  a  fixed  eighth  point. 

*  Three  algebraic  surfaces  whose  equations  are  of  degrees  m,  ?i,  p,  respectively, 
intersect  in  at  least  mnp  distinct  or  coincident  points.  If  they  have  more  than 
mnp  points  in  common,  then  they  have  one  or  more  curves  in  common.  For  a 
proof  of  this  theorem  see  Salmon:  Lessons  Introductory  to  Modern  Higher 
Algebra,  Arts.  73,  78.     We  shall  assume  the  truth  of  this  theorem. 


168  LINEAR  SYSTEMS   OF  QUADRICS        [Chap.  XI. 

These  points  are  called  eight  associated  points.  If  the  coordi- 
nates of  any  fixed  arbitrarily  chosen  point  {y)  are  substituted  in 
(8),  the  condition  that  {y)  lies  on  the  cpiadric  furnishes  one  linear 
relation  among  the  A^.  Hence  through  {y)  pass  all  the  quadrics 
of  a  pencil  and  therefore  a  proper  or  composite  quartic  curve 
lying  on  every  quadric  of  the  pencil.  This  quartic  curve  passes 
through  the  eight  associated  points  of  the  bundle. 

If  (?/)  is  chosen  on  the  line  joining  any  two  of  the  eight  asso- 
ciated points,  every  quadric  of  the  pencil  passing  through  it  will 
contain  the  whole  line,  since  each  quadric  of  the  pencil  contains 
three  points  on  the  line  (Art.  65,  Th.  II).  The  residual  intersec- 
tion is  a  proper  or  composite  cubic  curve  passing  through  the 
other  six  of  the  associated  points  and  cutting  the  given  line  in 
two  points. 

137.  Representation  of  the  quadrics  of  a  bundle  by  points  of  a 
plane.  Let  Ai,  \o,  A3  be  regarded  as  the  coordinates  of  a  point  in  a 
plane,  which  we  shall  call  the  A-plane.  To  each  point  of  the  A- 
plane  corresponds  a  definite  set  of  values  of  the  ratios  Aj  :  A2 :  A3  and 
hence  a  definite  quadric  of  the  bundle  (1)  and  conversely,  so  that 
the  quadrics  of  the  bundle  and  the  points  of  the  A-plane  are  in  one 
to  one  correspondence.  To  the  points  of  any  straight  line  in  the 
A-plane  correspond  the  quadrics  of  a  pencil  contained  in  the  bundle. 
The  line  wdll  be  said  to  correspond  to  the  pencil.  Since  any  two 
lines  intersect  in  a  point,  it  follows  that  any  two  pencils  of  quadrics 
contained  in  the  bundle  have  one  quadric  in  common. 

138.  Singular  quadrics  of  the  bundle.  Those  values  of  Aj,  A2,  A3 
which  satisfy  the  equation 

lAia,.  +  X2&a  +  V.J  =  0  (9) 

will  define  singular  quadrics  of  the  bundle.  Unless  special  rela- 
tions exist  among  the  coefficients  a^^,  b^^,  c^^,  none  of  these  cones 
will  be  composite,  for  in  that  case  all  of  the  first  minors  of  (9) 
must  vanish,  thus  giving  rise  to  three  independent  conditions  among 
the  A,,  A2,  A3,  which  are  not  satisfied  for  arbitrary  values  of  the 
coefficients.  It  follows  further  that,  under  the  same  conditions,  no 
two  cones  contained  in  the  bundle  have  the  same  vertex.  For,  if 
/r=  0,  L  =  0  were  two  cones  having  the  same  vertex,  then  every 


Arts.  136-139]  PLANE   SECTION   OF  A  BUNDLE  169 

cone  of  the  pencil  Ai7v'+ A2L  =  0  would  have  this  point  for  a  ver- 
tex. By  choosing  this  point  as  vertex  (0,  0,  0,  1)  of  the  tetrahe- 
dron of  reference,  the  pencil  could  be  expressed  in  terras  of  the 
three  variables  x^,  Xo,  x^.  Tlie  discriminant  of  this  pencil  equated 
to  zero  would  be  a  cubic  in  Ai  :  Aj  whose  roots  define  composite 
cones  which  were  shown  above  not  to  exist  for  arbitrary  values  of 

^iki   ^iki   ^ik- 

It  follows  from  (9)  that  the  points  in  the  A-plane  determined  by 
values  of  A],  Ao,  A3  which  define  cones  of  the  bundle  of  (8)  lie  on  a 
quartic  curve  C4.  Every  point  of  this  curve  defines  a  cone  of  the 
bundle,  and  conversely.  Each  cone  has  a  vertex,  and  it  was  just 
shown  that  no  two  cones  have  the  same  vertex.  We  have  therefore 
the  following  theorem  : 

Theorem.  The  vertices  of  the  cones  in  a  general  bundle  describe 
a  space  curve  J.  The  points  of  J  are  in  one  to  one  correspondence 
with  the  points  of  the  curve  G^  in  the  X-plane. 

The  four  points  in  which  any  line  in  the  A-plane  intersects  C4 
correspond  to  the  four  singular  quadrics  of  the  pencil  which  cor- 
responds to  the  line.  If  P  is  any  point  on  the  quartic  curve,  the 
tangent  line  to  C4  at  P  defines  a  pencil  of  quadrics  in  which  one 
singular  quadric  is  counted  twice ;  if  the  residual  points  of  inter- 
section of  the  tangent  line  and  C4  are  distinct  from  each  other  and 
from  the  point  of  contact,  the  characteristic  of  the  pencil  is  [211]. 
All  the  quadrics  of  the  pencil  pass  through  the  vertex  of  the  cone 
corresponding  to  the  point  of  contact. 

139.  Intersection  of  the  bundle  by  a  plane.  If  the  quadrics  of 
the  bundle  (8)  are  not  all  singular,  the  equation 


=  0,  (10) 


•wherein  s.^  =  X^a^t.  -f  XJI\^  +  X^c-^,  is  called  the  equation  of  the 
bundle  in  plane  coordinates.  If  the  coordinates  of  a  given  plane 
(a)  are  substantiated  in  (10),  the  resulting  equation,  if  it  does  not 
vanish  identically,  is  homogeneous  of  degree  three  in  Ai,  A2,  A3  and 


^11 

S12 

■Sl3 

^U 

Wl 

§12 

6'o2 

•*''23 

S,, 

«2 

Sl3 

^23 

■''33 

•%4 

^h 

Su 

S24 

^34 

S44 

U, 

"1 

^'2 

«3 

"4 

0 

170  LINEAR  SYSTEMS   OF  QUADRICS        [Chap.  XI. 

is  consequently  the  equation  of  a  cubic  curve  C3  in  the  A-plane. 
Equation  (10)  is  the  condition  that  the  section  of  the  quadric 
(Ai,  A2,  A3)  by  the  plane  (»)  shall  be  composite.  Every  such  com- 
posite conic  in  the  plane  («)  has  at  least  one  double  point.  It  will 
now  be  shown  that  the  locus  of  the  point  of  tangency  to  (?t)  of  the 
quadrics  of  the  bundle  which  are  touched  by  (11)  is  a  cubic  curve. 

The  equation  of  any  plane  (w)  may  be  reduced  to  x*4  =  0  by  a 
suitable  choice  of  coordinates.  Let  Ai,  A2,  A3  be  any  set  of  values 
of  Aj,  A2,  A3  which  satisfy  (10)  when  we  have  replaced  «i,  Wj,  M3,  each, 
by  zero  and  M4  by  1. 

The  section  of  the  quadric  Ai^l  + A2-B  + A3C=0  by  the  plane 
a;4  =  0  is  a  composite  conic  having  at  least  one  double  point  (^y^,  y^, 
2/3,  0).     The  coordinates  of  (jj)  must  satisfy  the  relations 

K^a,,y,  +  ~X.^h,^,  +  AaSc.,,?/,  =  0,  for  i  =  1,  2,  3. 

If  from  these  three  equations  Aj,  Ao,  A3  are  eliminated,  the  result  is 
the  equation  of  the  locus  of  the  point  of  contact  (?/).  Since  the  re- 
sulting equation  is  of  degree  three  in  the  homogeneous  variables 
.Vu  .%)  .V31  the  locus  is  a  cubic  curve.  It  is  called  the  Jacobian  of 
the  net  of  conies  in  the  given  plane. 

140.  The  vertex  locus  /.  The  order  of  a  space  curve  is  defined 
as  the  number  of  its  (real  and  imaginary)  intersections  with  a 
given  plane. 

We  shall  now  prove  the  following  theorem : 

Theorem.     The  vertex  locus  J  of  a  general  bundle  is  of  order  six. 

For,  the  condition  that  the  vertex  of  a  cone  of  the  bundle  lies 
in  a  given  plane  (»)  is  that  the  corresponding  point  in  the  A-plane 
lies  on  each  of  the  curves  (9)  and  (10).  The  theorem  will  follow 
if  it  is  shown  that  these  curves  have  contact  of  just  the  first  order 
at  each  of  the  common  points  so  that  their  twelve  intersections 
coincide  in  pairs. 

Let  the  given  plane  be  taken  as  x^  =  0.  The  equation  of  a  cone 
of  the  bundle  having  its  vertex  in  this  plane  Dan  be  reduced  to 

X,'  +  x-'  +  .r/  =  0, 
and  that  of  the  bundle  to  the  form 

X,A  -f  A^B  +  A3(.«-o'  +  .^3^  +  .V)  =  0. 


Arts.  139-141]        POLAR  THEORY  IN  A  BUNDLE 


171 


The  point  in  the  A-plane  corresponding  to  the  cone  is  (0,  0,  1). 
It  lies  on  C^^X)  and  on  C3(X).  It  is  to  be  shown  that  Ci(X),  Cs{\) 
have  the  same  tangent  at  (0,  0,  1),  but  that  they  do  not  have  con- 
tact of  higher  than  the  first  order.  In  (9)  put  c^^  =  %  =  C44  =  1 
and  all  the  other  c^^  =  0,  and  develop  in  powers  of  A3.  The  re- 
sult may  be  written  in  the  form 


(aiiAi  +  511X2)  V  + 


<A. 


+ 


<^13 
<^33 


I    1 4>n      <f>U 

^14        <^44 


0, 


wherein  <f>,^  =  a^^Xi  +  b.^X^  =  <l>ki- 

Similarly   in   (10)    put   Wi  =  ^2  =  ^3  =  0,   mJ  =  1,  c.jj.  =  0,   and 
develop  in  powers  of  A3.     The  result  is 


(auXi  +  bnX^)Xs''  + 


<^12 
<^22 


+ 


<^13 


<^13 
<^33 


Xs   + 


0. 


These  curves  both  pass  through  the  point  (0,  0,  1)  and  have  the 
same  tangent  anAi  +  &11X2  =  0  at  that  point.  By  making  the  two 
equations  simultaneous,  it  is  seen  that  they  do  not  have  contact 
of  order  higher  than  the  first  unless  anXi  -\-  611X2  is  a  factor  of 

011<^44   -    4>\i^, 

which  is  not  the   case   unless   particular  relations   exist  among  the 
coefficients  a.t,  ?>,.. 


141.    Polar  theory  in  a  bundle. 

Theorem.     T7ie  polar  planes  of  a  point  (y)  toith  regard  to  all  the 
guadrics  of  a  bundle  pass  through  a  fixed  point  {y'). 

For,  the  polar  plane  of  the  point  (?/)  with  regard  to  a  quadric 
of  the  bundle  Aj^  +  A2B  +  A3C  =  0  has  the  equation 

X^^a^t^x-y^  +  X^'^bii.x^y^  +  X^tcufc-y^  =  0. 

For  all  values  of  A,,  Ao,  A3  this  plane  passes  through  the  point  (?/') 
of  intersection  of  the  three  planes 

SffifT^y,  =  0,     ^b,,x,y,  =  0,     2c,iX,?/4  =  0.  (11) 

From  the  theorem  that  if  the  polar  plane  of  (y)  passes  through 
(y'),  then  the  polar  plane  of  (?/')  passes  through  (y),  it  follows  that 
all  the  points  in  space  are  arranged  in  pairs  of  points  (y),  (y') 


172 


LINEAR  SYSTEMS  OF  QUADRICS        [Chap.  XI. 


conjugate  as  to  every  quadric  of  the  bundle.     Since  the  coordinates 
t/i,  y^,  yz,  2/4  a^wd  v'u  y'2)  l/zi  y\  appear  symmetrically  in  the  equations 

2a.,/.2/A.  =  0,     ^h,,y\y,  =  0,     2c.y,?/,  =  0 

defining  the  correspondence  between  {y)  and  {y'),  the  correspond- 
once  is  called  involutorial. 

By  solving  the  equations  defining  the  correspondence  for  y\, 
y'2,  y'z,  y\  we  obtain 

2a2t.V*     2a3,j/^     ^o.^yk 


o■y^ 


^Kyk 


^^Akyk 


and  similar  expressions  for  y'^,  y\.  y\.  If  we  denote  the  second 
members  of  the  respective  equations  by  ^i{y),  then  replace  both 
2/j  and  y\  by  x^  and  x\,  respectively,  the  equations  defining  the 
involution  may  be  written  in  the  form 

a.<.  =  «/,,(.!;),     px,  =  <i>,{x').  (12) 

If  {y)  describes  a  plane  S'ti.i'j  =  0,  the  equation  of  the  locus  of 
(?/')  may  be  obtained  by  eliminating  the  coordinates  of  {y)  from 
(11)  and  the  equation  2'<j/,  =  0.     The  result  is 


1/1 

U2 

"3 

^a^k^k 

^^2k^k 

^Chk^k 

^K^k 

'^K^k 

2&3i^* 

2Ci,.Tfc 

^^2k^k 

Scsi-r^ 

=  0. 


(13) 


Hence,  if  (y)  describes  a  plane,  (?/')  describes  a  cubic  surface. 
Similarly,  if  (y')  describes  a  plane,  (y)  describes  a  cubic  surface. 

If  (?/')  describes  a  line  I,  the  point  (y)  to  which  it  corresponds 
describes  a  curve  of  order  three.  For,  corresponding  to  each 
intersection  of  the  locus  of  (y)  with  the  plane  2m,.x-=0  there  is 
a  point  of  intersection  of  I  and  the  cubic  surface,  image  of  the 
plane.  But  I  intersects  the  surface  (13)  in  three  points,  hence 
2?/,x\  =  0  intersects  the  locus  of  (y)  in  three  points ;  that  is,  the 
locus  is  a  curve  of  order  three.  Similarly,  if  (y)  describes  a 
straight  line,  (y')  will  describe  a  curve  of  order  three. 

The  vertex  locus  ,/  lies  on  the  surface  (13)  for  all  positions 
of  the  plane  '^n-x^  =  0.  For,  let  (?/')  be  any  point  on  J.  Since 
[y')  is  the  vertex  of  a  cone  belonging  to  the  bundle,  its  polar  plane 


Arts.  141,  142]         SOME   SPECIAL  BUNDLES  173 

with  respect  to  this  cone  is  indeterminate  (Art.  121).  Hence 
there  exists  a  set  of  values  of  A,,  As,  A3,  not  all  zero,  for  which  this 
plane  is  indeterminate.     It  follows  that  the  matrix 

'Xuj.o;,     •$a.,x^     2a3,.r,     la^X;^ 
'^bi,x^     :^b,,x^     2^3,.?-,     lb^,x^ 

St'li-^V         ^C2^X^         ■n<'zk''''jc         ^'^ik-^k 

is  of  rank  at  most  two.  Thus,  in  the  equation  of  the  cubic  sur 
face  (13),  the  coefficient  of  each  n-  vanishes  when  the  coordinates 
of  any  point  J  are  substituted  in  it ;  hence  the  equation  is  satisfied 
for  all  values  of  (?/j,  U2,  it^,  u^). 

Any  two  planes  '^u-x-  =  0,  2f .cc^  =  0  intersect  in  a  line;  their 
image  surfaces  intersect  in  a  composite  curve  of  order  nine,  consist- 
ing of  t/and  the  cubic  curve,  image  of  the  line.  If  the  point  (y) 
is  the  vertex  of  a  cone  belonging  to  the  bundle,  the  three  polar 
planes  of  (?/)  determined  by  (11)  belong  to  a  pencil.  Let  I  be  the 
axis  of  this  pencil.  Every  point  of  the  line  I  corresponds  to  (y) 
in  the  correspondence  (11),  since  it  is  involutorial. 

As  (y)  describes  J,  its  corresponding  line  I  describes  a  ruled 
surface  R.  The  image  of  a  cubic  surface  2?/;*^,  =0  in  the  involu- 
tion (12)  is  the  plane  'S.u-t/-  =  0  and  a  residual  surface  of  order 
eight.  As  this  residual  surface  is  the  locus  of  Z,  we  conclude 
that  the  ruled  surface  R  is  of  order  eight. 

142.  Some  special  bundles.  While  it  would  lead  beyond  the 
scope  of  this  book  to  give  a  complete  classification  of  bundles  of 
quadrics,  like  that  for  pencils  of  quadrics  as  developed  in 
Arts.  131-133,  still  it  is  desirable  to  mention  a  few  particular 
cases.  It  was  seen  (Art.  138)  that  in  the  general  bundle  there 
are  no  composite  quadrics.  But  bundles  containing  composite 
quadrics  may  be  constructed ;  for  example,  the  bundle 

Ai^  -f  A2S  +  X^x^x.  =  0 

evidently  contains  the  composite  quadric  ic,.T2  =  0.  If  Xj  =  0  inter- 
sects the  curve  of  intersection  of  ^  =  0,  5  =  0  in  four  points,  and 
if  ^2  =  0  intersects  it  in  four  points,  so  that  no  component  of  the 
curve  lies  in  either  plane  a*,  =  0,  ccj  =  0,  then  these  two  sets  of-four 
points  constitute  eight  associated  points. 


174  LINEAR  SYSTEMS  OF  QUADRICS        [Chap.  XI.. 

Every  point  of  the  line  a;i  =  0,  iC2  =  0  is  a  vertex  of  a  com- 
posite cone  of  the  bundle.  The  locus  J  consists  of  this  line  and 
of  a  residual  curve  of  order  five.  The  image  curve  Ci{X)  in  the 
A-plane  has  a  double  point  corresponding  to  the  composite  quadric, 
as  may  be  seen  as  follows.  The  equation  of  Ci{X)=0  now  has 
the  form 

V<^2('^-1)    ^2)  +  ^3<^3('^l>    -^2)  +  ^4(^1)    ^2)  =  0, 

in  which  <^2)  ^3;  ^4  do  not  contain  A3.  Hence  the  point  Ai  =  0, 
Aj  =  0  is  a  double  point  on  Ci{\)=  0 ;  it  corresponds  to  the  quadric 
a;iiC2  =  0.  The  points  of  Ci{X)  are  now  in  one  to  one  correspond- 
ence with  the  curve  of  order  five,  forming  one  part  of  J,  and  the 
double  point  is  associated  with  the  whole  line  x^^  =  0,  0^2  =  0. 

Similarly,  buudles  of  quadrics  may  be  constructed  having  eight 
associated  double  points  lying  on  two,  three,  four,  five,  or  six 
pairs  of  planes.  In  the  last  case  the  equation  of  the  bundle  may 
be  written  in  the  form 

AiC^i^  -  ^4')  +  K{^2'-  -  3^4')  +  K{x,^  -  re/)  =  0. 

The  eight  associated  points  are  (±1,  ±1,  ±1,  1).  The  curve 
J  consists  of  the  six  edges  of  a  tetrahedron  and  C4(A)  is  composed 
of  the  four  sides  of  a  quadrilateral.     Its  equation  is 

AiA2A3(Ai  +  A2  +  A3)=0. 

In  this  case  the  equations  (12)  of  the  involution  (1/),  (2/')  have  the 
simple  form 

y\  =  -,  1  =  1,2,3,4, 

in  which  o-  is  constant. 

Bundles  of  quadrics  exist  having  a  common  curve  and  one  or 
more  distinct  common  points.  The  spheres  through  two  fixed 
points  furnish  an  example. 

EXERCISES 

1.  Show  that  (1,  0,  0,  0),  (0,  1,  0,  0),  (0,  0,  1,  0),  (0,  0,  0,  1),  (1,  1,  1,  1), 
(1,  1',  -1,  -1),  (1,  -1,  1,  -1),  (1,  -1,  -1,  1)  are  eight  associated 
points. 


Arts.  142-144]    THE  JACOBIAN  SURFACE   OF  A  WEB       175 

2.  Prove  that  if  P  is  a  given  point  and  I  a  given  line  through  it,  there 
is  one  and  only  one  quadric  of  the  bundle  to  which  I  is  tangent  at  P. 

3.  Determine  the  characteristic  of  the  pencil  of  quadrics  in  a  general 
bundle  corresponding  to  : 

(o)  A  tangent  to  Ci(\). 

(6)    A  double  tangent  to  Ci(\). 

(c)   An  inflexional  tangent  to  Ci{\). 

4.  What  is  the  general  condition  under  which  C4(X)  may  have  a  double 
point  ? 

5.  Determine  the  nature  of  the  bundle 

Xi(a:i2  -  x.xz)  +  Mx^^  +  x^^  +  X3'  -  4  X42)  +X3(xi2 -  3:32)  =  0 
and  of  the  involution  of  corresponding  points  (y),  (?/'). 

6.  If  three  quadrics  have  a  common  self-polar  tetrahedron,  the  twenty-four 
tangent  planes  at  their  eight  intersections  all  touch  a  quadric. 

7.  Write  the  equation  of  a  bundle  of  quadrics  passing  through  two  given 
skew  lines  and  a  given  point. 

8.  If  four  of  the  eight  common  tangent  planes  of  three  quadrics  meet  in  a 
point,  the  other  four  all  meet  in  a  point. 

9.  Show  that  the  cubic  curve,  image  of  an  arbitrary  line,  intersects  the 
locus  of  vertices  J  in  8  points. 

10.    Show  that  the  surface  B  of  Art.  141  contains  J"  as  a  threefold  curve. 

143.  "Webs  of  quadrics.  If  J.  =  2o,ta;,.T^  =  0,  B  =  '^bn^x^x^  =  0, 
C  =  2c,fc.r,a-4  =  0,  i)  =  Sc^i^cc.a^^  =  0  are  four  quadrics  not  belonging 
to  the  same  bundle,  the  linear  system 

X,A  +  X^B  +  X3C+\,D  =  0  (14) 

is  called  a  web  of  quadrics.  Through  any  point  in  space  pass  all 
the  quadrics  of  a  bundle  belonging  to  the  web,  through  any  two 
independent  points  a  pencil,  and  through  any  three  independent 
points,  a  single  quadric  of  the  web. 

144.  The  Jacobian  surface  of  a  web.  The  polar  planes  of  a 
point  {y)  with  regard  to  the  quadrics  of  a  web  form  a  linear  system 

^i^^ik^iVk  +  K'^^\k^^yk  +  AaSCi^a^i?/,  4-  XiMik^iVk  =  0.         (15) 

If  the  point  (?/)  is  chosen  arbitrarily,  this  plane  may,  by  giving  Aj, 
A2,  X3,  X4  suitable  values,  be  made  to  coincide  with  any  plane  in 


176 


LINEAR  SYSTEMS   OF  QUADRICS        [Chap.  XI 


space,  unless  there  are  particular  relations  among  the  coefficients 
a^k,  6,-^.,  c.jt,  dij.  Thus  an  arbitrary  plane  is  the  polar  plane  of  (y) 
with  regard  to  some  quadric  of  the  web.  There  exists  a  locus  of 
points  (y)  whose  polar  planes  with  regard  to  all  the  quadrics  of  a 
web  pass  through  a  fixed  point  (?/').  This  locus  is  called  the 
Jacobian  of  the  web.  Since  the  equations  connecting  (y)  and  (?/') 
are  symmetrical,  it  follows  that  (y')  also  lies  on  the  Jacobian.  A 
pair  of  points  (?/),  (y')  such  that  all  the  polar  planes  of  each  pass 
through  the  other  are  called  conjugate  points  on  the  Jacobian. 

To  determine  the  equation  of  the  Jacobian,  we  impose  the  con- 
dition that  the  four  polar  planes  of  (y) 

pass  through  a  point. 


K.= 


It.     The  result 

is 

2au?/i     2a2i?/i 

S^aJ/i 

2a4i2/i 

:s&ii//i    2?>2i.Vi 

^b,i!/i 

^KVi 

2c,i?/,.      Scjii'/i 

'^c,,y, 

^CAiVi 

2rfu-,'/i       Srfoi?/; 

'^^hiVi 

2t?4i?/i 

=  0. 


(16) 


The  condition  that  a  point  (,;/)  is  the  vertex  of  a  cone  contained 
in  the  web  is  that  its  coordinates  satisfy  the  equations 

K%a,,y,  +  X,-%h,,y,  +  AaSc.,^/,  +  X,-^d,,y,  =0,  k=  1,  2,  3,  4     (17) 

for  some  values  of  A„  A2,  A.3,  A4. 

By  eliminating  A,,  A2,  A3,  A4  we  obtain  equation  (16).     This  gives 
the  theorem  : 

Theorem  I.     The  Jacobian  surface  is  the  locus  of  the  vertices  of  the 
cones  contained  in  the  web  of  quadrics. 

If  from  equations  (17)  we  eliminate  yi,  y^,  2/3,  2/4  we  obtain 


nx)- 


^u 

tn 

tn 

tu 

t2X 

^22 

t<a 

tu 

tsi 

^32 

tiS 

tz. 

tn 

<42 

t,-i 

tu 

=  0, 


in  which  tik  =  XiOifc  +  ^-^hik  +  X^Cik  +  "Xdik  =  tki.  Any  set  of  values 
of  Xi,  X2,  X3,  X4  for  which  ^(X)  =  0  determines  a  singular  quadric  of 
the  web.  Conversely,  the  parameters  of  every  singular  quadric  in 
(14)  satisfy  T(X)  =  0. 


Arts.  144-146]        WEB  WITH   SIX  BASIS  POINTS  177 

Since  T(\)  is  a  sjonmetric  determinant  there  are  ten  sets  of  values 
of  Xi,  X2,  Xs,  X4  for  which  it  is  of  rank  two.*  The  ten  corresponding 
quadrics  are  composite  and  each  line  of  vertices  lies  on  Kt  =  0,  hence 
we  have  the  theorem: 

Theorem  II.  The  Jacohian  of  the  general  web  of  quadrics  contains 
ten  lines. 

145.  Correspondence  with  the  planes  of  space.  The  polar  plane 
of  a  fixed  point  (y)  with  regard  to  any  quadric  Q  of  the  web  will 
be  called  the  associated  plane  of  (y)  as  to  Q.  When  Q  describes 
a  pencil,  its  associated  plane  will  describe  a  pencil ;  when  Q  de- 
scribes a  bundle,  its  associated  plane  will  describe  a  bundle.  The 
quartic  curve  of  intersection  of  two  quadrics  of  the  web  corre- 
sponds to  the- line  of  intersection  of  their  associated  planes,  and  to 
every  set  of  eight  associated  points  of  a  bundle  of  quadrics  in  the 
web  corresponds  one  point,  the  vertex  of  the  bundle  of  associated 
planes.  Through  any  two  points  a  straight  line  can  be  drawn, 
hence  through  any  two  sets  of  eight  associated  points  within  the 
web  can  be  passed  a  pencil  of  quadrics  belonging  to  the  web.  Since 
through  any  three  points  a  plane  can  be  passed,  it  follows  that  a 
quadric  of  the  web  can  be  foimd  which  passes  through  any  three 
sets  of  eight  associated  points  in  the  web. 

146.  Web  with  six  basis  points.  The  maximum  number  of  dis- 
tinct basis  points  a  web  can  have  without  having  a  basis  curve  is  six. 
Let  1,  2,  3,  4,  5,  6  designate  the  six  basis  points  of  a  web  having 
six  basis  points.  All  the  quadrics  of  the  web  through  an  arbitrary 
point  P  belong  to  a  bundle,  and  hence  have  eight  associated  points 
(Art.  136)  in  common,  the  eighth  point  P'  being  fixed  when  1,  2, 
3,  4,  5,  6  and  P  are  given.  Between  P  =  ($)  and  P'  =  (^')  exists 
a  non-linear  correspondence. 

We  shall  now  prove  the  following  theorem : 

Theorem  I.  In  the  case  of  a  tveb  ivith  six  distinct  basis  points, 
the  Jacobian  surface  Ki  =  0  is  also  the  locus  of  points  (|)  such  that 

*  Salmon:  Lessons  Introductory  to  Modem  Higher  Algebra,  Lesson  XIX. 
The  configuration  of  the  lines  on  the  Jacobian  has  been  studied  by  Reye.  See 
Crelle's  Journal,  Vol.  86  (1880). 


178  LINEAR  SYSTEMS   OP  QUADRICS        [Chap.  XI. 

In  order  to  prove  this  we  shall  prove  the  following  theorems : 

Theorem  II.  Tlie  quadrics  of  a  bundle  of  the  weh  which  pass 
through  the  vertex  of  a  given  cone  of  the  web  have,  at  this  vertex,  a 
common  tangent  line. 

Theorem  III.  Conversely,  if  all  the  quadrics  of  a  bundle  have 
a  common  tangent  line  at  a  given  point,  a  cone  beloyiging  to  the 
bundle  has  its  vertex  at  the  point. 

To  prove  Theorem  II,  let  the  vertex  of  the  given  cone  be 
(1,  0,  0,  0),  so  that  its  equation  (7=0  does  not  contain  x^.  Let 
^  =  0,  5  =  0  be  any  two  non-singular  quadrics  of  the  bundle 
passing  through  the  point,  so  that  a,,  =  0,  b^^  =  0.  The  equation 
of  the  tangent  plane  to  the  quadric  Ai^l -f  XnB  +  ^.3(7=  0  at 
(1,  0,  0,  0)  is 

'^•i(«i2^'2  +  «i3^3  +  «iA)  +  A2(&12.^'2  +  b^^x^  +  b^^x^  =  0. 

But  these  planes  all  contain  the  line 

a,2.T2  +  a^^x^  +  a^^x^  =  0,    b^^x^  +  b^^x^  +  b^x^  =  0, 

which  proves  the  proposition. 

To  prove  Theorem  III,  let  x-,  =  0,  0^2  =  0  be  the  equations  of  the 
line,  and  (0,  0,  0,  1)  the  common  point.     We  may  then  take 

A  =  2  a^^x^x^  +  <^(a7i,  x.^,  x^)  =  0, 
JB  =  2  624^22^4  +  i^Ca^'i,  x^,  Xi)=Q, 

C    =  w   CnX]X^  -f-  w  C243J2''^4     \    J  V**'!?    "^ii   "^3/^^  ") 

wherein  <^,  i/', /contain  only  x^,  x^,  Xj. 
In  the  bundle 

the  quadric  corresponding  to  Ai  =  —  0,4624,  A2  =  —  0,14024,  A3  =  0,4624 
is  a  cone  with  vertex  at  (0,  0,  0, 1)  since  the  equation  of  the  quadric 
does  not  contain  x^. 

Since  at  the  vertex  of  every  cone  two  associated  points  coin- 
cide, and  conversely,  at  every  coincidence  is  the  vertex  of  a  cone, 
the  proposition  of  Theorem  I  follows. 

The  ten  pairs  of  planes  determined  by  the  six  basis  points 
1,  2,  3,  4,  5,  6  taken  in  groups  of  three,  as,  for  example,  the  pair 


Art.  146] 


WEB  WITH   SIX  BASIS  POINTS 


179 


tXj^»<>2»t3*t'^ 

a^i 

«1 

1 

2    3    4     1 

0^2 

02 

1 

3    4     1'' 

Xj 

"a 

1 

^4123 

X, 

04 

1 

of  planes  (123),  (45G),  are  composite  quadrics  of  the  web.  The 
line  of  vertices  of  each  pair  lies  on  7t4  =  0.  The  surface  K^  =  0 
also  contains  the  fifteen  lines  joining  the  basis  points  by  twos, 
since  through  any  point  of  such  a  line  five  lines  can  be  drawn  to 
the  six  basis  points,  and  a  quadric  cone  of  the  web  is  fixed  by 
these  five  lines. 

If  the  basis  points  are  taken  for  vertices  of  the  tetrahedron  of 
reference,  the  unit  point,  and  the  point  (a^,  a^,  a^,  at),  the  equation 
of  K^  =  0  is  found  to  be 


=  0. 


This  surface  is  known  as  the  Weddle  surface.* 

If  in  (17)  the  values  of  y^,  y^,  y^,  y^  are  eliminated,  the  resulting 
equation  A(A)  =  0  of  degree  four  in  the  A^  will  define  those  values 
for  which  the  equation  XiA  +  X2B-\-X^C-{-X4D  =  0  is  a  cone  of 
the  web.  The  vertex  of  this  cone  is  a  point  (^)  =  (^').  Let  A,,  X.,, 
A3,  A4  be  considered  as  the  tetrahedral  coordinates  of  a  plane.  To 
each  plane  (A)  corresponds  a  quadric  of  the  web  (14)  and  con- 
versely. A  linear  equation  with  given  coefficients  aXi  +  bXn  + 
cA3+(/A4  =  0  determines  a  point  in  the  A-space  (Art.  91).  By 
making  this  equation  and  (14)  simultaneous,  we  define  a  bundle 
whose  basis  points  are  the  points  (x)  whose  coordinates  satisfy 
the  equations 

abed 

Of  the  eight  associated  points  so  determined,  the  given  points 
1,  2,  3,  4,  5,  6  are  six.  Either  of  the  remaining  points  P  =  ($), 
P'  =  ($')  will  uniquely  determine  the  other  and  also  uniquely 
determine  the  point  (a,  b,  c,  d)  in  the  A-space.  The  equation 
aXi  +  bXi  +  CA3  +  dXi  =  0  thus  defines  a  one  to  two  correspondence 
between  the  points  of  the  A-space  and  the  points  P  and  P.     For 


*  First  discussed  in  the  Cambridge  and  Dublin  Mathematical  Journal,  Vol.  5 
(1850),  p.  69. 


180 


LINEAR  SYSTEMS  OF  QUADRICS        [Chap.  XL 


points  of  A",  P  and  P'  coincide.     The  locus  of  the  corresponding 
point  (a,  h,  c,  d)  is  called  the  Kummer  surface.* 
We  have  thus  proved  the  following  theorem  : 

Theorem  IV.      Tlie  j^oints  of  the  Weddle  surface  and  the  points 
of  the  Kummer  surface  are  in  one  to  one  correspondence. 


EXERCISES 

1.  Show  that  all  the  quadrics  having  a  common  self-polar  tetrahedron 
form  a  web. 

2.  Determine  the  Jacobian  of  the  web  of  Ex.  1. 

3.  Determine  under  what  conditions  the  Jacobian  of  a  web  will  have  a 
plane  as  component. 

4.  Find  the  Jacobian  of  the  web  defined  by  the  spheres  passing  through 
the  origin  x  =  0,  ?/  =  0,  z  =  0. 

5.  Show  that  the  Jacobian  of  a  web  having  two  basis  lines  is  inde- 
terminate. 

6.  Discuss  the  involution  of  conjugate  points  (y),  (y')  for  the  web  of 
Ex.4. 

7.  Show  that  the  spheres  cutting  a  given  sphere  orthogonally  define  a 
web. 

8.  Show  that  the  equation  of  the  quadric  determined  by  the  lines  joining 
the  points  (1,  0,  0,  0),  (ai,  aj,  «3,  ^4);  (0,  1,  0,  0),  (0,  0,  1,  0);  (1,  1,  1,  1), 
(0,  0,  0,  1)  is 

x^Xl{a2  —  az)+  (a^T^  —  02X3)  +  Xi(aiXi  —  aiX2)  =  0. 

147.    Linear  systems  of  rank  r.     The  linear  system  of  quadrics 

A.i^i  +  X,Fo  +  ■'■-\-KF,  =  0,  (19) 

wherein 

is  said  to  be  of  rank  r,  if  the  matrix 


(20) 


"11 

"22 

"33      • 

"34 

n    "-2) 

"22 

"33        • 

.  a   <2) 
U.34 

"11 

a.,2<'' 

"33 

U34 

*  First  discussed  by  E.  E.  Kummer  in  the  Monatsberichte  der  k.  preussischen 
Akademie  der  Wissensehaften,  Berlin,  1863. 


Arts.  146-149]  APOLARITY  181 

is  of  rank  r,  that  is,  if  there  does  not  exist  a  set  of  values  of  Aj, 
A2,  •••,  Xrf  ^lot  all  zero,  such  that  the  expression 
k,F,  +  X,F^  +  :-  +  X,F, 

is  identically  zero.  All  the  quadrics  in  space  form  a  linear  sys- 
tem of  rank  ten,  since  the  equation  of  any  quadric  may  be  ex- 
pressed linearly  in  terms  of  the  ten  quadrics,  x^,  x^,  •••,  x^Xi  for 
which  the  matrix  (20)  is  of  rank  ten. 

All  the  quadrics  in  space  whose  coefficients  satisfy  10  —  r 
independent  homogeneous  linear  equations  form  a  linear  system 
of  rank  r.  For,  if  ^b-^x^x^  =  0  is  the  equation  of  any  quadric 
whose  coefficients  satisfy  the  given  conditions,  then  all  the  co- 
efficients 6j^  can  be  expressed  linearly  in  terms  of  the  coefficients 
of  r  quadrics  belonging  to  the  system.     Thus 

bi,  =  Aia.,<"  +  Vf.,"'  +  -  +  KaJ'\     h  k  =  1,  2,  3,  4,       (21) 

wherein  ^     ,,>  ^  „     .  ^ 

are  fixed  quadrics  belonging  to  the  system. 

Conversely,  10  —  r  independent  homogeneous  linear  conditions 
may  be  found  which  are  satisfied  by  the  coefficients  in  the  equa- 
tions of  the  quadrics  F,  =  0,  i^2  =  0,  -•,  F,  =  0,  and  consequently 
by  the  coefficients  in  the  equations  of  all  the  quadrics  of  the 
linear  system  (19)  of  rank  r. 

148.  Linear  systems  of  rank  r  in  plane  coordinates.  The  system 
of  quadrics 

AA  +  A.2<^2  +  •••  +A,$,  =  0, 

wherein  $,  =  2/3./"". "«  is  called  a  linear  system  of  rank  r  in 
plane  coordinates  if  there  does  not  exist  a  set  of  values  Ai, 
A2,  — ,  A^  for  which  the  given  equation  is  satisfied  identically. 
These  systems  may  be  discussed  in  the  same  manner  as  that 
considered  in  the  preceding  article. 

149.  Apolarity.  Let  F=  Sa-^x.x-^  =  0  be  the  equation  of  a 
quadric  in  point  coordinates  and  ^  ~  S/?,;,?/,?^^.  =  0  be  the  equation 
of  a  quadric  in  plane  coordinates.     If  the  equation 

2a,i^,,  ~  a„/3„  +  a,,f3,,  +  a,,l3,,  +  a,,(3»  +  2  a^^As  +  2  a,,l3,,  +  2  a,,^,, 
+  2  cu,(3,,  +  2  ao,A4  +  2  a,,/3,,  =  0  (22) 


182  LINEAR  SYSTEMS  OF  QUADRICS        [Chap.  XI. 

is  satisfied  by  the  coefficients  in  the  eqnations  of  the  two  quadrics, 
F=0  is  said  to  be  apolar  to  <!>=  0,  and  $  =  0  is  said  to  be  apolar 
to  F  =  0.  It  should  be  noticed  that  in  this  definition  the  equa- 
tion F=0  is  given  in  point  coordinates,  and  that  of  <I>  =  0  in 
plane  coordinates.  It  should  also  be  noticed  that  if  i^=0  and 
4>  =  0  are  two  given  apolar  quadrics,  and  if  2«i4.?/,%  =  0  is  the 
equation  of  i^  =  0  in  plane  coordinates,  and  '!S,bi;^x-x^  =  0  is  the 
equation  of  ^  =  0  in  point  coordinates,  then  it  does  not  necessarily 
follow  that  Sa.At  =  ^  because  SttaiSit  — "  0- 

In   order   to   show  the   significance  of  the   condition   (22)  of 
apolarity,  we  shall  prove  the  following  theorem : 

Theorem  I.     TJie  expression  a^^/Sik  is  a  relative  invariant. 

Let  the  coordinates  of  space  be  subjected  to  the  linear  trans- 
formation 

»,.  =  ttax'i  +  Ui^x'z  +  a^^^x'i  +  a^x'^,     i  =  1,  2,  3,  4 

of  determinant  T^O.  The  coordinates  of  the  planes  of  space 
undergo  the  transformation  (Art.  97) 

M.  =  Ai^n\  +  yl.ou'z  +  As^'s  +  Ai^i'i,     i  =  1,  2,  3,  4. 

The  equation  F(x)=  0  goes  into  '^^a'^^x'-x'^  =  0, 
wherein  (Art.  104) 

and  $  =  0  is  transformed  in  ^(i'-f.u-u\  =  0,  wherein 

The  proof  of  the  theorem  consists  in  showing  (Art.  104)  that 

In  the  first  member,  replace  a',^,  fi\^  by  their  values  from  the 
above  equations,  and  collect  the  coefficients  of  any  term  tti^/Si^  in 
the  result.     We  find 

hence 

which  proves  the  proposition. 

The  vanishing  of   this   relative  invariant  may  be  interpreted 
geometrically  by  means  of  the  following  theorem  : 


Art.  149]  APOLARITY  183 

Theorem  II.  If  F  =  0,  ^  =  0  are  apolar  quadrics,  there  exists 
a  tetrahedron  self-polar  as  <o  <l>  =  0  and  inscribed  in  F  =  0. 

This  theorem  should  be  replaced  by  others  in  the  following 
exceptional  cases  in  which  no  such  tetrahedron  exists. 

(a)  If  2^=0  is  a  plane  counted  twice.  In  this  case  (22) 
is  the  condition  that  the  coordinates  in  this  plane  satisfy 
$  =  0. 

(b)  If  $  =  0  is  the  equation  of  the  tangent  planes  to  a  proper 
conic  C  and  ii  F  =  0  intersects  the  plane  of  C  =  0  in  a  line 
counted  twice,  (22)  is  the  condition  that  this  line  touches  C. 

We  shall  consider  first  the  special  cases  (a)  and  (b). 

Let  F  =  (uiXi  +  »2'^2  +  "3^3  +  ^^^^^y^ 

Then  a^^  =  M,?/fc  and  (22)  reduces  at  once  to  $  =  0. 

In  case  (6),  let  the  plane  of  C  be  taken  as  ^4  =  0  and  the  line 
of  intersection  of  F  =0  with  X4  =  0  be  taken  as  x^  =  x^  =  0. 
Then 

$  =  fin<  +  PlM.^  +  /833"3'  +  2  /3i27(,«2  +  2  ^23^2^3  +  2  P,,U,U,  =  (i, 

and  F  =  a^^x^-  -\-  2  a^iXyX^  +  2  a^^x.x^  +  2  a^^x^Xi  +  2  a^^x^  =  0, 

where  a^  ^  0.  Hence  (22)  reduces  to  ^^  =  0,  that  is,  to  the  con- 
dition that  Xj  =  .i'4  =  0  touches  C. 

To  prove  Theorem  II,  excluding  cases  (a)  and  (6),  we  must 
consider  various  cases.  First  suppose  4>  =  0  is  non-singular. 
Choose  a  point  Pj  on  F  =  0,  not  on  the  intersection  F  =  0,  ^  —  0, 
and  find  its  polar  plane  tt,  as  to  4>  =  0.  In  tti  take  a  point  Pg  ^^^ 
F  =  0,  not  on  4>  =  0,  and  find  its  polar  plane  tt.^  as  to  ^  =  0.  On 
the  line  ttittj  choose  a  point  P^  on  F  =  0,  not  on  $  =  0,  and  find 
its  polar  plane  ttj.  If  the  point  of  intersection  of  ttj,  ttj,  ttj  is 
called  P4,  then  P1P2P3P4  =  TTiTToTTzTr^  is  taken  for  the  tetrahedron 
of  reference;  we  may,  by  proper  choice  of  the  unit  plane,  reduce 
the  equation  of  $  =  0  to  ^^i^  +  n.,^  +  11  ^^  -\-  ii^-  =  0.  Equation  (22) 
now  has  the  form  On  -f  «22  +  ^33  +  f'«  =  ^-  Since  three  of  the 
vertices  P,,  P2,  P3  were  chosen  on  P  =  0,  three  coefficients  a^,  =  0, 
hence  the  fourth  must  also  vanish,  which  proves  the  proposition 
for  this  case. 

It  should  be  observed   that  if   P  =  0,  *  =  0  define  the  same 


184  LINEAR  SYSTEMS   OF  QUADRICS        [Chap.  XI. 

quadvic,  equation  (22)  cannot  be  satisfied  since  tlieir  equations 
may  be  reduced  simultaneously  to 

F  =  x^^  +  x.^  +  xi  +  0^4^  =  0,         4>  =  u^  +  W2'  +  ^i  +  n^  =  0. 

Now  let  $  =  0  be  the  equation  of  the  tangent  planes  to  a 
proper  conic  c.     Take  the  plane  of  O  as  x^  =  0,  so  that 

)8h  =  )8,,  =  /334  =  1844  =  0. 

If  2^=0  is  composite  and  x^  is  one  component,  equation  (22) 
is  identically  satisfied.  In  this  case  we  may  take  three  vertices 
of  a  triangle  in  x^^^  self-polar  as  to  the  conic  C  and  any  point 
on  i^  =  0  not  on  .T4  =  0  as  vertices  of  a  tetrahedron  self-polar  to 
$  =  0  and  inscribed  in  jP  =  0.  If  jp'  =  0  consists  of  ^'4  =  0 
counted  twice,  (22)  expresses  the  condition  that  the  plane 
belongs  to  $  =  0,  whether  $  =  0  is  singular  or  not.  This  is  the 
exceptional  case  (a). 

If  CC4  =  0  is  not  a  component  of  /^  =  0,  (22)  has  the  form 

ttiiiSn  +  atSit  +  033/833  +  2  a,,^i.3  -f  2  «,3;8i3  +  2  a.^S1^=  0, 
which  is  the  condition  that  the  section  C"  of  i^  =  0  by  the  plane 
0:4  =  0  is  apolar  to  C. 

It  follows  by  the  theorem  for  apolar  conies  analogous  to 
Theorem  II  that  a  triangle  exists  which  is  inscribed  in  C"  and  is 
self-polar  to  C.  A  tetrahedron  having  the  vertices  of  this  tri- 
angle for  three  of  its  vertices  and  a  fourth  vertex  on  i^  =  0  but 
not  on  0^4  =  0  satisfies  the  condition  of  the  theorem  (dual  of 
Th.  I,  Art.  121). 

If  <I>  =  0  is  the  equation  of  two  distinct  points,  (22)  expresses 
the  condition  that  these  points  are  conjugate  as  to  ii^=  0.  This 
is  also  the  condition  that  a  tetrahedron  exists  which  is  inscribed 
in  F=  0  and  is  self-polar  to  <I>  =  0.  If  4>  =  0  is  the  equation  of 
a  point  counted  twice,  (22)  expresses  that  the  point  lies  on 
i^=  0.     This  is  the  dual  of  the  exceptional  case  (a). 

In  each  of  the  above  cases,  the  teti-ahedron  which  satisfies  the 
conditions  of  the  theorem  can  be  chosen  in  an  infinite  number  of 
ways,  hence  we  have  the  following  theorem. 

Theorem  III.  If  one,  tetrahedron  exists  ivhich  is  inscribed  in 
F=  0  and  is  self-polar  as  to  $  =  0,  the^i  an  infinite  number  of  such 
f£trahedra  exist. 


Art.  149]  APOLARITY  185 

By  duality  we  have  the  following  theorems : 

Theorem  IV.  //'  i^  =  0,  $  =  0  are  apolar  quadrics,  there  exists 
a  tetrahedron  self-polar  as  to  F  =  0  and  circumscribed  <o  4>  =  0. 

Theorem  V.  If  one  tetrahedron  exists  which  is  circumscribed  to 
$  =  0  and  is  self-polar  as  to  F  =  0,  then  an  infinite  number  of  such 
tetrahedra  exist. 

Moreover,  both  the  exceptional  cases  of  Theorem  II  have  an 
immediate  dual  interpretation:  they  will  not  be  considered  further. 

With  the  aid  of  these  results  we  can  now  give  an  interpretation 
to  the  vanishing  of  the  coefficients  0  and  ©'  of  equation  (3),  Art. 
124,  and  of  %{u),  %{n)  of  equation  (7),  Art.  135.  If  5  =  0  in  (1) 
is  non-singular,  let  its  equation  in  plane  coordinates  be  2;8i^.tt,i<^.  =  0. 
Since  /S.-^  is  the  first  minor  of  6-^  in  the  discriminant  of  B  =  0,  it 
follows  at  once  from  equation  (3)  that  ©'  =  'S.ai^^i^-  Hence  0'  =  0 
is  the  condition  that  ^  =  0  is  apolar  to  B  =  0.  If  5  =  0  is  a  cone, 
it  is  similarly  seen  that  0'  =  0  is  the  condition  that  the  vertex  of 
the  cone  B  =  0  lies  on  ^  =  0.  If  5  =  0  is  composite,  0'  is  iden- 
tically zero,  independently  of  A,  since  the  discriminant  of  2?  =  0 
is  of  rank  two,  hence  all  the  coefficients  ^^^  vanish.  An  analogous 
discussion  holds  for  0  =  0. 

The  surface  *i(w)  =  0  (Art.  135)  may  be  defined  as  the  envelope 
of  a  plane  which  intersects  ^4  =  0  in  a  conic  which  is  apolar  to 
the  conic  in  which  it  intersects  B  =  0.  For  particular  singular 
quadrics  this  definition  will  not  always  apply. 

Let  an  arbitrary  plane  of  *i(w)  =  0  be  taken  as  x^  =  0.  It  fol- 
lows from  equation  (7)  that 

I  «11&22&33  1    +    1  ^ll«22&33  |    +    |  hAi^hz  \    =  0.  (23) 

Let  the  sections  of  A  =  0,  B  =  0  by  x^  =  0  he  C,  C,  respectively. 
If  C"  is  not  composite,  it  is  seen  by  writing  the  equation  of  C  in 
line  coordinates  that  (23)  is  the  condition  that  C  is  apolar  to  C. 
If  C  is  a  pair  of  distinct  lines,  (23)  is  the  condition  that  their 
point  of  intersection  lies  on  C.  If  C  is  a  line  counted  twice,  (23) 
is  satisfied  identically  for  all  values  of  a^^,  since  all  the  first  minors 
of  the  discriminant  of  C"  vanish. 

An  analogous  discussion  holds  for  *2(^)  =  ^- 


186  LINEAR  SYSTEMS  OF  QUADRICS        [Chap.  XI. 

150.  Linear  systems  of  apolar  quadrics.  Since  equation  (22)  is 
linear  in  the  coefficients  of  i^=  0,  from  Art.  147  we  may  state  the 
following  theorem : 

Theorem  1.  All  the  quadrics  apolar  to  a  given  quadric  form  a 
linear  system  of  rank  nine. 

Conversely,  since  the  coefficients  of  the  equations  of  all  the 
quadrics  of  a  linear  system  of  rank  nine  satisfy  a  linear  condition 
which  may  be  written  in  the  form  of  equation  (27),  we  have  the 
further  theorem  : 

Theorem  11.  All  the  quadrics  of  any  linear  system  of  rank  nine 
are  apolar  to  a  fixed  quadric. 

From  the  condition  that  a  plane  counted  twice  is  apolar  to  a 
quadric  (Art.  149),  it  follows  that  this  fixed  quadric  is  the  envelope 
of  the  double  planes  of  the  given  linear  system. 

If  a  quadric  F  —0  is  apolar  to  each  of  r  quadrics 

4.1  =  2A.,<»M,%  =  0,     $,  =  2y8,,<^>.^,%  =  0,  ..., 
<!>,  =  :^/3J'-\r,,  =  0, 

the  coefficients  in  its  equation  satisfy  the  r  conditions 

It  follows  that  if  a  quadric  is  apolar  to  each  of  the  given  quadrics, 
it  is  apolar  to  all  the  quadrics  of  the  linear  system 

The  conditions  that  this  linear  system  is  of  rank  r  are  equivalent 
to  the  conditions  that  the  corresponding  equations  (24)  are  inde- 
dendent.     Hence : 

Theorem  111.  All  the  quadrics  apolar  to  the  quadrics  of  a  linear 
syste^n  of  rank  r  in  plane  coordinates  form  a  linear  system  of  rank 
10  —  r  in  point  coordinates  and  dually. 

EXERCISES 

1.  Find  the  equation  of  the  quadric  in  plane  coordinates  to  which  all  the 
quadrics  through  a  point  are  apolar. 


Art.  150]    LINEAR  SYSTEMS  OF  APOLAR  QUADRICS       187 

2.  How  many  double  planes  are  there  in  a  general  linear  system  of  rank 
seven  in  point  coordinates  ? 

3.  Show  that  all  the  pairs  of  points  in  a  linear  system  of  rank  six  in  plane 
coordinates  lie  on  a  quartic  surface. 

4.  Show  that  all  the  spheres  in  space  form  a  linear  system  and  find  its 
rank. 

5.  Find  the  system  apolar  to  the  system  in  Ex.  4. 

6.  Show  that  a  system  of  confocal  quadrics  (Art.  84)  is  a  linear  system  of 
rank  two  in  plane  coordinates.  Detei-mine  the  characteristic  and  the  singular 
quadrics  of  the  system  (Art.  133). 

7.  Show  that,  if  the  matrix  (20)  is  of  rank  r'  <  r,  the  system  of  quadrics 
(19)  is  a  linear  system  of  rank  r'. 


CHAPTER   XII 

TRANSFORMATIONS  OF  SPACE 

151.  Projective  metric.  In  order  to  characterize  a  transfor- 
mation of  motion,  either  translation,  or  rotation,  or  both,  or  a  trans- 
formation involving  motion  and  reflection,  as  a  special  case  of  a 
projective  transformation,  it  will  first  be  shown  under  what  cir- 
cumstances orthogonality  is  preserved  when  a  new  system  of 
coordinates  is  chosen. 

If  the  new  axes  can  be  obtained  from  the  old  ones  by  motion 
and  reflection,  the  plane  t  =  0  must  evidently  remain  fixed,  and 
the  expression  x"^  + 1/"^  +  z^,  which  defines  the  square  of  the  dis- 
tance from  the  point  (0,  0,  0,  1)  to  the  point  (x,  y,  z,  1),  must  be 
transformed  into  itself  or  into  {x  —  atf  +  (y  —  hty  +  (^  —  cty, 
according  as  the  point  (0,  0,  0, 1)  remains  fixed  or  is  transformed 
into  the  point  (a,  h,  c,  1).  It  will  be  shown  that,  conversely,  any 
linear  transformation  having  this  property  is  a  motion  or  a  motion 
and  a  reflection. 

152.  Pole  and  polar  as  to  the  absolute.  We  shall  first  point  out 
the  following  relation  between  the  direction  cosines  of  a  line  and  the 
coordinates  of  the  point  in  which  it  pierces  the  plane  at  infinity. 

Theorem  I.  T7ie  homogeneous  coordinates  of  the  point  in  ivhich 
a  line  meets  the  plane  at  infinity  are  proportional  to  the  direction 
cosines  of  the  line. 

The  equations  of  a  line  through  the  given  finite  point  (xq,  y^,  Zq,  t^ 
and  having  the  direction  cosines  (A,  [x,  v)  are 

t{<ix  —  X(f,  __  tf^y      y^t  __  t^  —  Zff  ^.tv 

A  /A  V 

The  point  {x,  y,  z,  0)  in  which  the  line  pierces  the  plane  at  infinity  is 
given  by  the  equations 

A        /x         V 
from  which  the  theorem  follows. 

188 


Arts.  151,  152]  POLE   AND   POLAR  189 

We  shall  now  establish  the  following  theorems  concerning  poles 
and  polars  as  to  the  absolute. 

Theorem  II.  The  necessary  and  sufficient  condition  that  a  plane 
and  a  line  are  perpendicular  is  that  the  line  at  infinity  in  the  jilane  is 
the  polar  of  the  point  at  infinity  on  the  line  as  to  the  absolute. 

The  absolute  was  defined  (Art.  49)  as  the  imaginary  circle  in  the 
plane  at  infinity  defined  by  the  equations 

x'-\-y''-\-z^  =  0,         t  =  0.  (2) 

The  polar  line  as  to  the  absolute  of  the  point  (A,  fi,  v,  0)  in  which 
the  line  (1)  intersects  the  plane  at  infinity  is 

Xx  +  fxy  +  v2  =  0,         «  =  0.  (3) 

The  equation  of  any  plane  through  this  line  is  of  the  form 

\x  +  tiy  +  vz-\-  kt  =  0.  (4) 

These  planes  are  all  perpendicular  to  the  line  (1).  Conversely, 
the  equation  of  any  plane  perpendicular  to  the  line  (1)  is  of  the 
form  (4) ;  the  plane  will  therefore  intersect  the  plane  at  infinity 
in  the  line  (3). 

Theorem  III.  T7ie  necessary  and  sxifficient  condition  that  two 
lines  are  perpendicular  is  that  their  jyoints  at  infinity  are  conjugate 
as  to  the  absolute. 

The  condition  that  two  lines  are  perpendicular  is  that  each  lies 
in  a  plane  perpendicular  to  the  other,  that  is,  that  each  intersects 
the  polar  line  of  the  point  at  infinity  on  the  other  as  to  the  absolute. 

Finally,  since  two  planes  are  perpendicular  if  each  contains  a 
line  perpendicular  to  the  other,  we  have  the  following  theorem  : 

Theorem  IV.  Tlie  necessary  and  sufficient  condition  that  two 
planes  are  perpendicular  is  that  their  lines  at  infinity  are  conjugate 
as  to  the  absolute. 

A  tangent  plane  to  the  absolute  is  conjugate  to  any  plane  pass- 
ing through  the  point  of  contact ;  in  particular,  it  is  conjugate  to 
itself.  It  should  be  observed  that  the  equation  of  a  tangent  plane 
to  the  absolute  cannot  be  reduced  to  the  normal  form,  hence  we 
cannot  speak  of  the  direction  cosines  of  such  a  plane. 


190  TRANSFORMATIONS   OF   SPACE        [Chap.  XII. 

Consider  the  pencil  of  planes  passing  through  any  real  line. 

We  may  choose  two  perpendicular  planes  of  the  pencil  as  x  =  0, 

y  =  0,  and  write  the  equation  of  any  other  plane  of  the  pencil  in 

the  form 

y  =  mx. 

The  equations  of  the  two  tangent  planes  to  the  absolute  which 
pass  through  this  line  are  y  =  ix  and  y  =  —ix.  By  using  the 
usual  formula  to  obtain  the  tangent  of  the  angle  <^  between  y  =  ix 
and  y  =  mx,  we  obtain 

.        .        m  —  i        m  —  i       1 

tan  (^  = = =  -  =  —  * 

1  +  im      i(m  —  i)      i 

independent  of  m.  For  this  reason  tangent  planes  to  the  absolute 
are  called  isotropic  planes.  The  cone  having  its  vertex  at  (a,  b,  c) 
and  passing  through  the  absolute  has  an  equation  of  the  form 

(x  -  ay  +  {y-  by  +  (z-  cy  =  0. 

If  we  employ  the  formula  of  Art.  4  for  the  distance  between 
two  points,  we  see  that  the  distance  of  any  point  of  the  cone  from 
ihe  vertex  is  equal  to  zero.  For  this  reason  the  cone  is  called  a 
minimal  cone.  Moreover,  if  Pj  and  Pj  are  any  two  points  on  the 
same  generator,  since 

VP,-VP,  =  P,P,, 

we  conclude  that  the  distance  between  any  two  points  on  any  line 
that  intersects  the  absolute  is  zero.  For  this  reason  these  lines 
are  called  minimal  lines.     They  have  no  direction  cosines  (Art.  3). 

153.  Equations  of  motion.  Let  an  arbitrary  point  P  be  referred 
to  a  rectangular  system  of  coordinates  x,  y,  z,  t  and  to  a  tetrahe- 
dral  system  ccj,  .Tj,  x^,  x^,  with  the  restriction  that  .T4  =  0  is  the  equa- 
tion of  the  plane  at  infinity  t  =  0.  The  equations  connecting  the 
two  systems  of  coordinates  are 

crx  =  Xx^  +  A'xj  +  A"iC3  +  hXi, 

ay  =  ixx^  +  fjjx^  +  ii"x^  4-  h'x^,  . 

,  (tz  =  vXi  +  v'x2  +  v"Xi  4-  h"Xi,  ^  ' 

at  =  X4. 
Divide  the  first  three  equations  of  (5)  by  the  last,  member  by 

member,  and  replace  the  non-homogeneous  coordinates  -,etc.,by 


Arts.  153,  154]     PROJECTIVE  TRANSFORMATIONS  191 

x\  y\  z'  and  ^S  etc.,  by  x\,  x\,  o:\.     ]f  P  is  any  point  not  in  the 

plane  at  infinity,  we  shall  prove  the  following  theorem  : 

Theorem  I.  Tlie  most  general  linear  transformations  of  the  form 
(5)  that  ivill  transform  the  exjn'ession 

a-'2  +  y'-^  +  2'2  i7ito  x',2  +  x'.,^  +  x-.^ 

are   the   rotations  and   reflections  about    the  j)oiiit  x'  =  0',  y'  =  0, 
^'  =  0. 

If  we  substitute  the  values  of  x',  y',  z'  in  the  expression 
x'^  4.  y'i  -I-  2;'2^  we  obtain 

(\x\  +  \'x',  +  X"x',  +  hy  +  (^x\  +  ix'x\_  +  fx:'x\  +  h'Y 

+  {yX\    +   v'x'.   +   v"x\+h"y. 

If  this  is  equal  to  x\^  +  x'^  +  x'^  for  all  finite  values  of  x\,  x\, 
X3,  we  have  the  following  relations 

X2  +  ^2  -}-  v2  =  X'2  +  fj-"  +  v''  =  X'"  +  H-"^  +  v'"  =  1, 

XX'  +  f,fx'  +  vv'  =  X'X"  +  fji'fji"  +  v'v"  =  X"X  +  /x'V  +  v"v  =  0,       (6) 

hx  +  h'fx  +  h"v  =  0,  hX'  +  AV  +  h"v'  =  0,  hX"  +  h'lx"  +  /i"v"=  0. 

Since  the  determinant  |  Xp-'v"  \  is  not  zei'O,  it  follows  that 
h  =  h'  =  h"  =  0.  The  formulas  (6)  which  do  not  contain  h,  h', 
h"  are  exactly  the  relations  among  the  coefficients  to  define  a 
rotation  or  a  rotation  and  reflection  about  the  origin  (Art.  37). 
This  proves  the  proposition. 

By  similar  reasoning  we  may  prove  the  theorem : 

Theorem  II.     Transformations  that  ivill  transform 

a;'2  +  ^'2  +  z'^  into  (x\  -  ay  +  (x'^  -  by  +  (x',  -  cy 
consist  of  motion  or  of  motion  and  reflection.^ 

154.  Classification  of  projective  transformations.  The  equations 
of  any  projective  transformation  (Art.  98)  are  of  the  form 

fCX  1  ^  Ctu^l  "T  <^12**'2  ~r  Ctl3'''3     I     0(l4'^4> 

kX  2  ^  02)3^1  4"  tt22'''2     1     ^"23*^3     I     '^24'''4>  xwv 

kX  3  =  0^31  a^i  +  <^32'*'2  "T  <^ 33*^3  "T  C34'^4> 

kX  4  =  C41-'^l  "T  ^42^*2    I     '^43'^3     I     tt44^4' 


192 


TRANSFORMATIONS  OF  SPACE        [Chap.  XII. 


We  shall  now  consider  the  problem  of  classifying  the  existing 
types  of  such  transformations  and  of  reducing  their  equations  to 
the  simplest  form. 

The  invariant  points  of  the  transformation  (7)  are  determined 
by  those  values  of  k  which  satisfy  the  equation 


D(k) 


,-k 

«12 

«13 

«14 

«21 

Ctoo   A- 

a23 

024 

«31 

«32 

a33  —  k 

«34 

041 

«42 

«43 

a. 

^-k 

=  0. 


(8) 


The  classification  will    depend   fundamentally  on    the    invariant 
factors  (Art.  125)  of  this  determinant. 

In  equation  (7),  (x)  and  {x')  are  regarded  as  different  points, 
referred  to  the  same  system  of  coordinates.  In  order  to  simplify 
the  equations,  we  shall  refer  both  points  to  a  new  system  of 
coordinates.  To  do  this  both  (x)  and  (x')  are  to  be  operated  upon 
by  the  same  transformation 

We  shall  use  the  symbols  {x),  (y)  to  indicate  coordinates  of  the 
same  point,  referred  to  two  different  systems  of  coordinates,  while 
equations  between  (.r)  and  {x')  or  between  (y)  and  (y')  will  define 
a  projective  transformation  between  two  different  points,  referred 
to  the  same  system  of  coordinates. 

Let  fcj  be  a  root  of  D{k)  =  0.     The  four  equations 

(«ii  —  ki)x^  +  a^.^^  +  ais^s  +  a^Xi  =  0, 

aoi-Ti  +(022  —  ''^■1)^2  +  023^3  +  ^ii^i  =  0> 
Osi^'l  +  a32^"l  +(033  —  ^"l)-^'3  +  034-»*4  =  0, 
041X1    +  042X2  4-  043^3   +(«44  —  K)Xi  =  0 

are  therefore  consistent  and  determine  at  least  one  point  invariant 
under  the  transformation. 


Let 


2/3.A.  =  0        {i  =  2,  3,  4) 


be  the  equations  of  three  planes  passing  through  this  invariant 
point  but  not  belonging  to  the  same  pencil,  and  let 


2/Su.T,  =  0 

4=1 


Art.  154]  PROJECTIVE   TRANSFORMATIONS  193 

be  the  equation  of  any  plane  not  passing  through  the  invariant 
point.     If  now  we  put 

y,  =  ^I3,,x„  i  =  1,  2,  3,  4, 

and  solve  the  equations  for  the  x-, 

Xi  =  2yit?/,„  and  put  also  x\  =  ^yiky'k, 
*=i 

then  substitute  these  valaes  in  the  members  of  (7),  the  new  equa- 
tions, when  solved  for  y\,  will  be  of  the  form 

y'l  =  Kyi  +  binJ/2  +  ^132/3  +  &i4y« 

y\  =  622^2  +  ^232/3  +  ^242/4. 

y'3  =         Kyi  +  Kyz  +  hiy» 

y\=  bi.y2-\-bi^y^  +  biiyi. 

Without  changing  the  vertex  (1,  0,  0,  0),  the  planes  3/2  =  0,  7/3  =  0, 
2/4  =  0  may  be  replaced  by  others  by  repeating  this  same  process 
on  the  last  three  equations;  in  this  way  we  may  replace  the 
coefficients  632,  ^42  by  0 ;  by  a  further  application  to  the  variables 
2/3>  2/4  we  may  replace  643  by  0. 

Referred  to  the  system  of  coordinates  just  found,  the  equations 
of  the  projective  transformation  (7)  are 

X  I  =  fC^Xi  -f-  C12X2  +  Ci3^3  +  ^'l4''^4> 

(9) 


x',= 

C22X2  T"  C23''C3  +  C24X4, 

x\  = 

^333^3  +  C34X4, 

x\  = 

CaaXa, 

in  which  C22,  C33,  C44  are  all  roots  of  D  (k)  =  0. 

Equations  (9)  represent  the  form  to  which  the  equations  of  any 
projective  transformation  may  be  reduced.  The  further  simplifi- 
cation depends  upon  the  values  of  the  coefficients,  that  is,  upon 
the  characteristic  (Art.  127)  of  D{k). 

If  c^i^  0  and  C33  ^  C44,  make  the  further  transformation 

^1  ^  2/ij    ^2  ==  2/2J     ^3  =  ^3  H         '      >    ^4  ^  y*- 

^44      ^33 

On  making  this  substitution  we  reduce  the  equations  of  (9)  to  a 
form  in  which  the  coefficient  C34  is  replaced  by  zero. 


194  TRANSFORMATIONS  OF   SPACE        [Chap.  XII. 

In  any  case,  if  i  <,k  and  c^^  ^  c^.^,  we  may  always  remove  the 
term  c,^  by  replacing  x\.  by  x^  -\ — ^''^'^^'     in  both  members  of  the 

equation.  If  c„  =  c^^.  and  c^^  ^0,  by  a  change  of  unit  point,  o.^ 
may  be  replaced  by  unity  ;  thus,  if  C33  =  C44  and  C34  4^  0,  by  writing 
034X4  =  1/4,  we  obtain  the  equations 

^'3  =  6332/3     +  ?/4, 

.V'4  =  C332/4. 

These  two  types  of  transformations  will  reduce  the  equations  to 
their  simplest  form  in  every  case  in  which  D{k)  =  0  has  no  root 
of  multiplicity  greater  than  two. 

If  D(k)  =  0  has  one  simple  root  k^  and  a  triple  root  k^,  the  pre- 
ceding method  can  be  applied  to  reduce  the  equations  of  the 
transformation  to  , 

X  1  =  K^Xi, 

X  2  =                 "'2*''2  i~  ^23*^3     I     ^24'''4J 

X  2=  KqX^  -f"  '^34*''4> 

X  4  ^^  K^X^t 

In  case  a24  =  0,  the  preceding  method  can  be  applied  again ;  thus, 
if  a34^0,  a^  9^0,  each  may  be  replaced  by  unity;  if  coeflBcients 
a23,  a24,  a34  are  zero,  the  transformation  is  already  expressed  in  its 
simplest  form.  If  a24  =  0,  either  or  both  of  the  coefficients  023 
and  a34,  if  not  zero,  may  be  replaced  by  unity  by  a  transformation 
of  the  type  just  discussed. 

If  O24  ^  0,  a34  ^  0,  replace  x^  by  the  substitution 

■     «->4?/3 
^34 

In  the  transformed  equation,  the  new  a24  is  zero.     In  the  same 

way,  if  0*24  ^  0,  ^,34  =  0,  but  aog  =^  0,  put 

«23 

and  the  same  result  will  be  accomplished.     Finally,  if  a24=?!=0, 

but  O34  =  0,   023  =  ^)   P"t 

^1  =  1/1,    x.,  =  y3,    X3  =  y2,    x^  =  y^  (10) 

in  both  members  of  the  equation.  Now  a24  =  0,  and  the  complete 
reduction  can  be  made  as  before. 


Arts.  154,  155]     FORMS  OF  TRANSFORMATIONS 


195 


If  D{k)=0  has  a  fourfold  root  k^,  equations  (9)  reduce  to 

'^1*^2    1     ^^23*^3     1     ^^24*^4? 


X  2  — 
iK  3  = 

x\  = 


rC^X^. 


By  transformations  analogous  to  those  in  the  preceding  case,  the 
coefficients  cii^,  a^i,  and  024  may  be  reduced  to  zero,  and  the  coeffi- 
cients ai2,  «23>  3'i^d  a^i  to  zero  or  to  unity. 

This  completes  the  problem  of  reduction.  The  determination 
of  the  locus  of  the  invariant  points  and  the  characteristic  of  D(k) 
in  the  various  cases  is  left  as  an  exercise  for  the  student.  The  re- 
sults are  collected  in  the  following  table. 


projective  transformations. 

Locus   OF    InVAKIANT    I'uINTS 

Four  distinct  points. 

Two   distinct,  two  co- 
incident points. 

Two     distinct     points 
and  a  line. 

x^    One      distinct,     three 
coincident  points. 

A  point  and  a  line. 

A  point  and  a  plane.     C 

Two   pairs    of    coinci- 
dent points. 

Two  coincident  points 
and  a  line. 

Two  lines. 
^    Four  coincident  points. 


155.    Standard  forms  of 

equations  of 

CHARACrERISTI.- 

EurATUiNs 

[1111] 

X  1  =:  K^Xi, 

Jj    •)    ^—    /I'Q'^S 

^  3  ^^  "^3**'3) 

tJC  ^  A/^X^ 

[112] 

X  J  ^  '*'l'^'l> 

*fc  0  —  A.  2**^2 

.7;  3  :=  /CjXj  -|-  ^4, 

ii'  4  ^  A'3X'4 

[11(11)] 

»!/  1  — ^  /i.  1  iC/ 1  • 

X  2  — ■  fi/^^^'Z 

^  3  ^=  "^3'^3) 

X  4  =:  ^'3a•4 

[13] 

X  1  =  fCiXi, 

X  2  —  ''-'2**^2    ' 

iC  3  =  K2X^  -p  iC^, 

X  4  ^^  tt'i'^A 

[1(21)] 

X ,  =  kiXi, 

X  2  ■—  /toXo 

a*  3  ^  /Cj^'s  4"  •'^4) 

X  4  —  /i'2**^4 

[1(111)] 

X  1  =  ft'i^i. 

X  2  —  "'2    2 

X  I  =:  fC^X^^ 

X^  2  ~~  ^2*^4 

[22] 

X  J  —  A/jU/j  -p  *^23 

X  2  —  KyOutj 

a;  3  =  K^^  -\-  x^, 

JO  4  —  /C2X4 

[2(11)] 

X  1  =  k^Xi  -\-  X2, 

X  2  """"  A/ 1X0 

a;  3  =  k.^x^y 

•C  4  —  ^-2X4 

[(11)(11)] 

X 1  =  k^Xi, 

•t/  2  ^~  *^\*'^2 

X  3  =  K2X3y 

*C  4  —  h-^JCA 

[4] 

x\  =  A'la-i  -f  X2, 

X  2  =  A.'iX*2  + 

a.  3  =  a'jiCs  -\-  x*4, 

X  4  -^  /tj^X4 

196 

TRANSFORMATIONS  0 

Oharactbristic 

Equatio 

N8 

[(22)] 

X  1  =  KiXi  +  X2, 

X  2  ^^^  nTj^2 

x\  =  k,X3  +  Xi, 

X  4  =  li^iX^ 

[(13)] 

X  i^  fCiX^, 

X  2  —  '*^i^2  ~i       I 

iC  3  =  riiX^  -\-  X^y 

X  4  =^  /tj.?/^ 

[(112)] 

X  1  =:  fC^Xi, 

*C  9  A/  J  »*/2 

a;  3  =  /C1X3  +  x^, 

X  ^  —  rtJjU/j 

[(1111)] 

X  1  =  fCiXi, 

iC  2  —  'i-i'^2 

iC  3  =  tCiX^f 

EXERCISES 

SPACE        [Chap.  XII, 

Locus  OF  Invariant  Points 

A  line. 
A  lina 

A  plane. 

All  points  of  space ; 
the  identical  trans- 
formation. 


1.  In  type  [1111]  obtain  the  necessary  and  sufficient  condition  that  the 
transformation  obtained  by  applying  the  given  transformation  p  times  is  the 
identity. 

2.  In  [1(111)]  show  that  the  line  joining  any  point  P  to  its  image  P' 
always  passes  through  the  invariant  point. 

3.  In  Ex.  2,  let  O  be  the  invariant  point,  and  let  a  line  PP'  intersect 
the  invariant  plane  in  3L  Show  that  the  cross  ratio  of  OMPP'  is  constant. 
This  transformation  is  called  perspectivity.  If  the  points  OMPP'  are  har- 
monic, it  is  called  central  involution. 

4.  In  [(11)  (11)]  show  that  the  line  joining  any  point  P  to  its  image  P' 
meets  both  invariant  lines,  and  that  the  cross  ratio  of  P,  P'  and  these  points 
of  intersection  is  constant. 

5.  Discuss  the  duals  of  the  types  of  transformations  of  Art.  155. 

156.  Birational  transformations.  Besides  the  projective  trans- 
formations, we  have  already  met  (Arts.  141,  146)  with  certain 
non-linear  transformations  in  which  corresponding  to  an  arbitrary- 
point  (x)  is  a  definite  point  (x')  and  conversely.  These  are  all 
particular  illustrations  of  a  class  of  ti*ansformations  which  will 
now  be  considered. 

Let 

x\  =  <f>i{x„  x^,  x^,  X,),     i  =  1,  2,  3,  4  (11) 

be  four  rational  integral  functions  of  Xi,  a;,,  X3,  x^,  all  of  the  same 
degree.  When  Xi,  x.,  x^,  x^  are  given,  the  values  of  x\,  x\_,  x:^,  a^i 
are  uniquely  determined,  hence  corresponding  to  a  point  (a;)  is  a 


Art.  156]         BIRATIONAL  TRANSFORMATIONS  197 

definite  point  (x').  If  the  equations  (11)  can  be  solved  rationally 
for  Xi,  X2,  X3,  x^  in  terms  of  x\,  x'o,  x\,  x' ^ 

X,  =  U^\,  x'„  x\,  x\),     i  =  1,  2,  3,  4,  (12) 

in  which  all  the  functions  1//^  are  of  the  same  degree,  then  to  a 
point  {x')  also  corresponds  a  definite  point  {x).  In  this  case  the 
transformation  defined  by  (11)  is  called  birational ;  that  defined  by 
(12)  is  called  the  inverse  of  that  defined  by  (11). 

When  the  point  (.«')  describes  the  plane  2«',«'i  =  0,  the  corre- 
sponding point  {x)  describes  the  surface 

w'i<^i(^)  +  w'2</>2(^)  +  u'3<f)-i{x)  +  u\<f>^(x)  =  0.  (13) 

This  surface  will  be  said  to  correspond  to  the  plane  (/<■').  If  the 
m',  are  thought  of  as  parameters,  we  may  say  :  corresponding  to  all 
the  planes  of  space  are  the  surfaces  of  a  web  defined  by  (13). 
In  the  same  way  it  is  seen  that,  corresponding  to  the  planes 
"Siii-x-  =0  of  the  system  (.«),  are  the  surfaces  of  the  web 

u,il,,(x')  +  ti.^lx')  +  n,i(;,(x')  +  u,^,(x')  =  0.  (14) 

Three  planes  («')  which  do  not  belong  to  a  pencil  have  one  and 
only  one  point  in  common,  henoe  three  surfaces  of  the  web  (13), 
which  do  not  belong  to  a  pencil,  determine  a  unique  point  (x) 
common  to  them  all,  whose  coordinates  are  functions  of  the  coor- 
dinates of  («'). 

This  fact  shows  that  in  the  case  of  non-linear  transformations 
the  web  defined  by  (13)  cannot  be  a  linear  combination  of  arbi- 
trary surfaces  of  given  degree.  For  if  the  <^-  are  non-linear,  any 
three  of  them  intersect  in  more  than  one  point,  but  it  was  just 
seen  that  of  the  points  of  intersection  there  is  just  one  point 
whose  coordinates  depend  upon  the  particular  surfaces  of  the  web 
chosen.  The  remaining  intersections  are  common  to  all  the  sur- 
faces of  the  web.  They  are  called  the  fundamental  points  of  the 
system  (x)  in  the  tranformation  (11).  When  the  coordinates  of  a 
fundamental  point  are  substituted  in  (9),  the  coordinates  of  the 
corresponding  point  (x')  all  vanish.  For  the  fundamental  points 
the  correspondence  is  not  one  to  one.  The  fundamental  points  of 
(x')  are  the  common  basis  points  of  the  surfaces  *pi(x')  =  0. 


198  TRANSFORMATIONS  OF   SPACE        [Chap.  XII. 

157.  Quadratic  transformations.  We  have  seen  (Art.  98)  that 
if  the  4>i  are  linear  functions,  the  transformation  (11)  is  projective, 
and  that  no  point  is  common  to  all  four  planes  <fy,(x)  =0.  The 
simplest  non-linear  transformations  are  those  in  which  the  cf>i  are 
quadratic.  We  shall  consider  the  case  in  which  all  the  quadrics 
of  the  web  have  a  conic  c  in  common. 

Let  the  equations  of  the  given  conic  be 

2i/.a5.-  =  0,         f{x)  =  0. 

Any  quadric  of  the  system 

2?/.a;.(AiaJi  +  ^^x^  +  X3X3  +  \^x^)  +  \5f(x)  =  0 

will  pass  through  this  conic.  Among  the  quadrics  of  this  system 
those  passing  through  an  arbitrary  point  P  define  a  web.  Any 
two  quadrics  Hi  =  0,  H.^^O  of  this  web  intersect  in  a  space 
curve  consisting  of  the  conic  c  and  a  second  conic  c'  which  passes 
through  P.  The  planes  of  c  and  of  c'  constitute  a  composite 
quadric  belonging  to  the  pencil  determined  by  i/i  =  0  and  H  =  0, 
and  the  conies  c,  c'  lie  on  every  quadric  of  the  pencil.  Hence  c,  c' 
intersect  in  two  points,  as  otherwise  the  line  of  intersection  of  the 
two  planes  would  have  at  least  three  points  on  every  quadric  of 
the  pencil,  which  is  impossible. 

Any  third  quadric  H^  =  0  of  the  web  but  not  of  the  pencil 
determined  by  H^  =  0,  H^  =  0  passes  through  c  and  P.  The  plane 
of  c'  intersects  H^^O  in  a  conic  c"  passing  through  P  and  the 
two  points  common  to  c,  c'  and  in  just  one  other  point.  The  posi- 
tion of  this  fourth  point  of  intersection  depends  on  the  choice  of 
the  bundle  H^  =  0,  //a  =  (^  H^  =  0.  We  have  thus  proved  that  the 
web  of  quadrics  defined  by  a  conic  and  a  point  P  has  the  neces- 
sary property  mentioned  in  Art.  156  possessed  by  the  web  deter- 
mined by  a  birational  transformation. 

Let  the  equations  of  the  conic  c  be 

x^  =  0,     e^x^  +  62^2"  +  ^3^3*  =  0. 

If  P  is  not  on  the  plane  x^  =  0,  it  may  be  chosen  as  vertex 
(0,  0,  0,  1)  of  the  tetrahedron  of  reference.  The  equation  of  the 
web  has  the  form 


Art.  157]  QUADRATIC   TRANSFORMATIONS  199 

In  analogy  with  equation  (11)  we  may  now  put 

ic  1  ^  ^iX^y  a;  2  =  X2X^,     x  ^  :=  XyC^,  x  ^^  ^i-^i    r  62-'K2  -r  63X3 .     \^0) 

The  most  general  form  of  the  transformation  of  this  type  may  be 
obtained  by  replacing  the  x'^  by  any  linear  functions  of  them  with 
non-vanishing  determinant. 

In  the  derivation  of  equations  (12)  it  makes  no  difference 
whether  the  conic  c  is  proper  or  composite,  hence  three  cases 
arise,  according  as  ej  =  62  =  ^s  =  1  or  ^j  =  ej  =  1,  63  =  0  or  e^  =  1, 
62  =  ^3  =  0.     The  equations  are 

Cb  1  ^^  •t/iX^^        Jb  o  —  JUo^iy        **^  3  —  **^3*^4        ^  4  •""  **^i     "t"  *^2       1"      3  *  \     / 

CC  1  — —  **^i"-'45        *^  0  — '  **^2    4)  3  —  •^3*^4        *^  4  — ~  **^l       1*  ^^2  •  \    ) 

CC  1  — —  "^1*^41        *^  2  —  *^2    4'  3  ^"~  **^3*^4        *^  4  —  *^l    •  \    / 

Now  let  P  approach  a  poiut  K  on  the  conic  c.  If  c  is  com- 
posite, suppose  its  factors  are  distinct  and  that  K  lies  on  only 
one  of  them.  In  the  limits  the  line  KP  is  tangent  to  all  the  quad- 
rics  of  the  web  determined  by  c  and  P.  But  the  tangent  to  c  at  K 
is  also  tangent  to  all  these  quadrics  at  K.  Hence  the  plane  of 
these  two  lines  is  a  common  tangent  plane  to  all  the  quadrics  of 
the  web  at  K=  P. 

Let  P  be  taken  as  (1,  0,  0,  0),  the  common  tangent  plane  at  P 
as  X2  =  0,  and  let  the  equations  of  the  conic  be  reduced  to  x^  =  0, 
iCia;2  +  ex^^  =  0.     The  equation  of  the  web  has  the  form 

KiX^Xi  +  X2X3X4  +  Agx/  +  \i{xiX2  +  6X3^)  =  0. 

The  two  cases,  according  as  e  =  1  or  e  =  0,  give  rise  to  the 
transformations 

X  J  =  ^20^4,    X  2  — ™  •^3^4,     *^  3  ^—  «^4  ,    •C  4  ^  ^\*^2  "I"  *^3  >  V     / 

•  1  "^  X^Xa*     X  2  —'  *^3**'4,     X  3  —  fcCj  ■     *C  4  —  XyX^  \     / 

of  this  type. 

Finally,  let  c  be  composite  and  let  the  point  K  which  P  ap- 
proaches lie  on  both  components  of  c.  Since  all  the  quadrics 
thi'ough  c  have  in  this  case  the  plane  of  c  for  common  tangent 
plane  at  K,  the  point  P  must  approach  c  in  such  a  way  that  the 
line  KP  approaches  the  plane  of  c  as  a  limiting  position.  The 
conies  in  which  the  quadrics  of  the  web  are  intersected  by  any 
plane  through  Pand  A"  have  two  points  in  common  at  K  and  one 


200  TRANSFORMATIONS  OF  SPACE        [Chap.  XII. 

at  P.  Hence  in  the  limit,  all  these  conies  must  have  three  inter- 
sections coincident  at  K  =  P. 

Let  the  equations  of  c  be  Xi  =  0,  x^  +  ex^  =  0,  and  the  coordi- 
nates of  P  be  (1,  0,  0,  0).  The  equations  of  the  system  of  rank 
five  of  quadrics  through  c  is 

XiX^Xi  +  \2X.Xi  +  XsX^Xi  +  XiXi^  +  X^(xi^  +  6X3^)  =  0. 

The  section  of  this  system  by  any  plane  through  P,  different  from 
.T4  =  0,  will  consist  of  a  system  of  conies  touching  each  other  at  P. 
The  required  web  belongs  to  this  system  and  satisfies  the  condi- 
tion that  its  section  by  any  plane  through  P  other  than  a'4  =  0  is  a 
system  of  conies  having  three  intersections  coincident  at  (1,  0,  0, 0). 
The  equations  of  the  section  by  the  plane  .T3  =  0  are 

Ai^jS/^  -f-  A2'1^2'^4  ~r  A4.T4   -|-  A5X2    ^  "5      '^3  ~^  ^• 

All  these  conies  touch  each  other  at  P.  Let  \\,  A'2,  A'4,  A'5  be  the 
parameters  of  one  conic,  and  Aj,  A2,  A4,  A5  of  another  contained  in 
this  system.  The  equations  of  the  lines  from  (1,  0,  0,  0)  to  the 
two  remaining  intersections  of  these  two  conic  are 

(AiA'a  -  A^A'O-bf  +(AiA'2  -  A2A'i)avr4  +(AiA'4  -  K^'i)^*'  =  0. 

One  of  these  remaining  points  is  also  at  P  if  AjA'j  —  A5A',  =  0. 
Hence  all  the  quadrics  of  the  web  satisfy  a  relation  of  the  form 
A5  -f  A;Ai  =  0.  It  is  no  restriction  to  put  A;  =  1.  It  can  now  be 
shown  that  the  conies  cut  from  the  quadrics  of  the  web  A5  +  Ai  =  0 
by  any  plane  aiX^  4-  aoXj  -f  O3.V3  =  0  through  P  have  three  coinci- 
dent points  in  common  at  P. 
The  equation  of  the  web  is 

Ai.r.,.r4  +  X.,XyXi  +  X:^Xi^  +  Xi(x2^  +  ex^  -  0^10:4)  =  0. 

The  two  birational  transformations  defined  by  webs  of  quadrics 
of  this  type  are 

X  1  =  X2.T4,     .1;  2  =  •'^3"^4»     '^  3  ^^  '^*4  >     3J  4  =  .^2    -f-  .^3  XyC^.  yj  ) 

X  ]  =  ■^2"^4»    '^  2  ^^  •^i'^i:    *^  3  ^  •''4  J    iK  4  =  3?2         X^X^.  \^yj 

The  inverse  transformations  of  forms  (a)  •••  (g)  are  also  quadratic. 
For  this  reason  transformations  of  this  type  are  called  quadratic- 
quadratic. 


Arts.  158,  159]  RECIPROCAL   RADII  201 

158.  Quadratic  inversion.  A  geometric  inpthod  of  constructing 
some  of  the  preceding  types  of  birational  transformations  will 
now  be  considered.  Given  a  quadric  A  and  a  point  0.  Let  P  be 
any  point  in  space,  and  P'  the  point  in  which  the  polar  plane  of 
P  as  to  A  cuts  the  line  OP.  The  transformation  defined  by  hav-  ' 
ing  P'  correspond  to  P  is  called  quadratic  inversion.  If  0  does 
not  lie  on  the  quadric  ^  =  0,  let  0  =  (0,  0,  0,  1)  and  let  the  equa- 
tion of  ^1  =  0  be 

If  P  =  (yi,  y^,  2/3,  y^,  the  coordinates  of  P'  are 

y\  =  yiyt,  y't  =  yiyi,  y'z  =  M4,  y\  =  euVi^  +  622/2^  +  e^x^^ 

which  include  forms  (a),  (6),  (c).     If  0  lies  on  A,  we  may  take 

A  =  x^-  +  e,x,^  -  x,x,  =  0,   0  =  (0,  0,  0,  1). 

The  coordinates  of  P'  in  this  case  are  functions  of  y^,  y^,  y^,  y^ 
defined  b}''  (/)  and  (r/).  The  quadratic-quadratic  transformations 
(a),  (b),  (c),  (/),  (g)  can  therefore  be  generated  in  this  manner. 

159.  Transformation  by  reciprocal  radii.     If,  for  the  quadric 
^  =  0  (Art.  158)  we  take  the  sphere 

a;2  ^y^  +  z^  =  m""  (16) 

and  for  0  the  center  of  this  sphere,  the  equations  of  the  trans- 
formation assume  the  form 

x'  =  k'^xt,  y'  =  k^yt,  z'  =  kht,  if  =  x^  +  y^ -^  x".  (17) 

On  account  of  the  relation 

OP '  OP'  =  ¥  (18) 

existing  between  the  segments  from  0  to  any  pair  of  correspond- 
ing points  P,  P',  it  is  called  the  transformation  by  reciprocal  radii. 
Any  plane  not  passing  through  0  goes  into  a  sphere  passing 
through  0  and  the  circle  in  which  the  given  plane  meets  the 
sphere  (16),  which  is  called  the  sphere  of  inversion. 

The   fundamental    elements    are   the   center   0,  the   plane   at 
infinity,  and  the  asymptotic  cone  of  the  sphere  of  inversion. 


202  TRANoFORMATIONS   OP   SPACE        [Chap.  XII. 

A  plane  ax  +  by -\- cz -\-  dt  =  0  not  passing  through  the  origin 
(d  =^  0)  is  transformed  into  a  sphere 

ak^xt  +  bkhjt  +  ckht  +  d(.«2  ^y'i+z'^)=0 

passing  through  the  origin. 

A  plane  passing  through  the  origin  is  transformed  into  a  com- 
posite sphere  consisting  of  the  given  plane  and  the  plane  at  in- 
finity. We  shall  say  that  planes  through  the  origin  are  trans- 
formed into  themselves. 

A  sphere 

a(a;-  +  y'' +  z"^) -\- 2  fxt  -f  2  gyt  -f  2  hzt  +  mf  =  0  (19) 

not  passing  through  the  origin   (in  -^^  0)  is  transformed  into  the 

sphere 

wi(a;2  +  2/2  +  z-")  +  2fk\xt  +  2  gk'^yt  +  2  hk^zt  -^  ak' =  0.      (20) 

The  factor  x^  +  y"^  +  z"^  can  be  removed  from  the  transformed 
equation.  A  sphere  passing  through  the  origin  {in  =  0)  is  trans- 
formed into  a  composite  sphere  consisting  of  a  plane  and  the 
plane  at  infinity. 

If  any  surface  passes  through  the  origin,  its  image  is  seen  to  be 
composite,  one  factor  being  the  plane  at  infinity.  The  plane  at 
infinity  is  the  image  of  the  center  0,  which  is  a  fundamental 
point. 

In  particular,  the  sphere  (19)  will  go  into  itself  if  m  =  ak^ ; 
but  this  is  exactly  the  condition  that  the  sphere  (19)  is  orthogonal 
to  the  sphere  of  inversion,  hence  we  may  say  : 

Theorem  I.  Tlie  sphe7-es  tchich  are  orthogonal  to  the  sphere  of  in- 
version go  into  themselces  when  transformed  by  reciprocal  radii. 

We  shall  now  prove  the  following  theorem : 

Theorem  II.  Angles  are  preserved  ivhen  transformed  by  recipro- 
cal radii. 

Let       AyX  +  B{y  +  C^z  +  D^t  =  0,  A^x  -f  B^y  +  G^z  +  0^1  =  0 

be  any  two  planes.  The  angle  6  at  which  they  intersect  is  de- 
fined by  the  formula  (Art.  15) 

QQg    Q  _  ^1^2  +  A-S2  +  C]  C2 ^  /o\\ 

-JIaJTW+^WaFVW+W) 


Arts.  159,  160]  CYCLIDES  203 

These  planes  go  into  the  spheres 

Z)i(x2  +  ?/2  -f  z2)  +  A^k'^xt  +  BJi-yt  +  CJiht  =  0, 
i>2(x2  +  ?/2  +  z2)  +  A.Ji'-xt  +  JSjA;^^^  4-  Cpzt  =  0. 

Since  the  angle  of  intersection  of  two  spheres  is  the  same  for 
every  point  of  their  curve  of  intersection  (Art.  51)  and  both 
spheres  pass  through  0,  we  may  determine  the  angle  at  which  the 
spheres  intersect  by  obtaining  the  angle  between  the  tangent 
planes  at  0.     These  tangent  planes  are 

A^x  +  B^y  +  CjZ  =  0,  A^x  +  B^y  +  C^z  =  0, 

hence  the  angle  between  them  is  defined  by  (21).  Since  the  angle 
of  intersection  of  any  two  surfaces  at  a  point  lying  on  both  is  de- 
fined as  the  angle  between  their  tangent  planes  at  this  common 
point,  the  proposition  is  proved. 

160.  Cyclides.  Since  lines  are  transformed  by  reciprocal  radii 
into  circles  passing  through  0,  a  ruled  surface  will  be  transformed 
into  a  surface  containing  an  infinite  number  of  circles.  A  quadric 
has  two  systems  of  lines,  hence  its  transform  will  contain  tw^o  sys- 
tems of  circles,  and  every  circle  of  each  system  will  pass  through 
0.  Moreover,  the  quadric  contains  six  systems  of  circular  sections 
lying  on  the  planes  of  six  parallel  pencils  (Art.  82).  Hence  the 
transform  will  also  contain  six  additional  systems  of  circles,  not 
passing  through  0,  but  so  situated  that  each  system  lies  on  a 
pencil  of  spheres  passing  through  0. 

By  rotating  the  axes  (Art.  37),  we  may  reduce  (Art.  70)  the 
equation  of  any  quadric  not  passing  through  0  to  the  form 

^  ax"  +  hy""  +  c2^  +  i^  +  2  Ixt  -f  2  myt  -f  2  nzt  =  0  (22) 

without  changing  the  form  of  the  equation  of  the  sphere  of  inver- 
sion.    By  transforming  this  surface  by  reciprocal  radii,  we  obtain 
(a;2  +  t/2  +  22)2  +  2  k^x""  +  ?/  +  z2)(/.i-  +  my  +  nz)t 
+  k\aj^  +  by^  +  cz'^)f  =  0. 
This  surface  is  called  the  nodal  cyclide ;  it  contains  the  absolute 
as  a  double  curve  and  has  a  double  point  at  the  point  0.* 

*  A  point  P  on  a  surface  is  called  a  double  point  or  node  when  every  line 
through  P  raeets  the  surface  in  two  coincident  points  at  P.  A  curve  on  a  surface 
is  called  a  double  curve  when  every  point  of  the  curve  is  a  double  point  of  the 
surface. 


204  TRANSFORMATIONS  OF   SPACE        [Chap.  XII. 

If  the  given  quadric  is  a  cone  with  vertex  at  P,  its  image  will 
have  a  double  point  at  0  and  another  at  P'.  The  circles  which 
are  the  images  of  the  generators  of  the  cone  pass  through  0 
and  P'. 

The  equation  of  the  cone  may  be  taken  as 

a{x-  fty -\-b{y-  gty  +  c(z  -  htf  =  0  (23) 

and  the  equation  of  the  transform  is 
(a/2  _,_  ^g2  _,_  e/i2)(a;2  4-  ^2  _,_  ^2^2  _  q  m{x^  +if  +z'){afx  +  bgy  +  chz) 
+  k\ax'  4-  &/  +  cz'y  =  0. 

This  surface  has  a  node  at  the  origin  and  at  the  transform 

(/)  9^  h,  p-\-  rf  -\-  7*2)  of  the  vertex  of  the  cone  (23).     It  is  called  a 

binodal  cyclide. 

If,  in  equation  (22),  6  =  c,  so  that  the  given  quadric  is  a  surface 
of  revolution,  the  transformed  equation  may  be  written  in  the 
form 

[ic2+  ^2  _,_  ^2  _|_  ^tQ,-^  +  my  +  nz)t  +  ^  IH'^'y  +  (a  -  h)k*xH'^ 


- k4lx 4- my  +  nz  +  ^  k''t\H''  =  0. 


It  has  a  node  at  0  and  at  the  points  in  which  the  line  a;  =  0, 
2lx-{-2  my  +  2  nz  +  k'^ht  =  0  intersects  the  sphere  a;^  +  3/^  +  2^  +  2  Ixt 
+  2  myt  -f  2  nzt  +  bkH-  =  0.     It  is  called  the  trinodal  cyclide. 

Finally,  if  the  cone  (21)  is  one  of  revolution,  the  resulting 
cyclide  has  four  nodes,  and  is  called  a  cyclide  of  Dupin.  If  the 
center  of  inversion  is  within  the  cone,  so  that  no  real  tangent 
planes  can  be  drawn  to  the  cone  through  the  line  OP,  the  surface 
is  called  a  spindle  cyclide;  if  the  center  is  outside  the  cone,  the 
resulting  surface  is  called  a  horn  cyclide. 

The  generating  circles  of  a  cone  of  revolution  intersect  the  recti- 
linear generators  at  right  angles.  Since  both  the  lines  and  the 
circles  are  transformed  into  circles  and  angles  are  preserved  by 
the  transformation,  we  have  the  following  theorem: 

Theorem  III.  Through  each  point  of  a  cyclide  of  Diqjin  pass 
two  circles  lying  entirely  on  the  surface.  Tliese  circles  meet  each 
other  at  right  angles. 


Art.  160]  CYCLIDES  205 

A  particular  case  of  the  spindle  cyclide  is  obtained  by  taking 
the  axis  of  the  cone  through  the  center  of  inversion.  The  result- 
ing cyclide  is  in  this  case  a  surface  of  revolution.  It  may  be 
generated  by  revolving  a  circle  about  one  of  its  secants.  If  the 
points  of  intersection  of  the  circle  and  the  secant  are  imaginary, 
the  cyclide  is  called  the  ring  cyclide.  It  has  the  form  of  an 
anchor  ring.  In  this  case  all  the  nodes  of  the  cyclide  are 
imaginary. 

EXERCISES 

1.  If  A  consists  of  a  pair  of  non-parallel  planes  and  0  is  taken  on  one  of 
them,  show  that  the  quadratic  inversion  reduces  to  the  linear  transformation 
defined  in  Art.  155,  Ex.  8  as  central  involution. 

2.  Obtain  the  transform  of  the  ellipsoid 

?!  +  m!  +  ^  =  1 

a?-      y-      €^ 
with  regard  to  the  sphere  x~  +  ;/'-  ■\-  z'  —  1.     How  many  systems  of  circles 
lie  on  the  resulting  surface  ?     Show  that  four  minimal  lines  pass  through  0 
and  lie  on  the  surface. 

3.  Show  that  the  transform  of  the  paraboloid  ax^  +  hxp-  =  2  ^  by  reciprocal 
radii  is  a  cubic  surface.  How  many  systems  of  circles  lie  on  this  surface  ? 
How  many  straight  lines  ? 

4.  Discuss  the  transform  of  a  quadric  cone  by  reciprocal  radii  when  the 
center  of  the  sphere  of  inversion  lies  on  the  surface  but  is  not  at  the  vertex. 

5.  Show  that  a  surface  of  degree  n  passing  k  times  through  the  center  of 
inversion  is  transformed  by  reciprocal  radii  into  a  surface  of  degree  2(n  —  A;),    2,  >- 
having  the  absolute  as  an  {n  —  ^•)-fold  curve. 

6.  Show  that  the  center  of  an  arbitrary  sphere  is  not  transformed  into 
the  center  of  the  transformed  sphere  by  reciprocal  radii. 

7.  Given  the  transformation 

X'l  =  {Xx  —  Xz)X-2,     x'o=(Xi  —  X2).r:i,     X'3  =  (Xl  —  X2).r4,     X'i  =  X2X3. 

Find  the  equations  of  the  inverse  transformation  and  discuss  the  basis  points 
in  (x). 

8.  Given  the  transformation 

x'l  =  X1X2,     X'o  =  X2X3,     X':?  =  .r^Xi,     X'i  =  X4(Xi  -I-  X-2  +  X3). 

Find  the  equations  of  the  inverse  transformation.     Discuss  the  basis  points 
in  the  web  of  quadrics  XiXiXo  +  X2X2X;i  +  \3X3X1  +  X4X4(xi  +  X2  +  X3)  =  0. 


CHAPTER   XIII 

CURVES  AND  SURFACES   IN   TETRAHEDRAL  COORDINATES 

I.   Algebraic  Surfaces 

161.    Number  of   constants   in  the  equation  of  a  surface.     The 

locus  of  the  equation 

f(^)  =  2    ,o^,g,  aa^Y6«'i"'»2 V^/  =  0,  (1) 

«!  p!  y!  6! 

wherein  a,  /S,  y,  8  are  positive  integers  (or  zero)  satisfying  the 

equation   «  +  /5+  y  +  8  =  n,   is   called   an   algebraic   surface   of 

degree  n. 

If  the  equation  is  arranged  in  powers  of  one  of  the  variables, 

as  .^4,  thus 

n^,"  +  u.x,'^-^  +  . . .  +  w„  =  0,  (2) 

in  which  u-  is  a  homogeneous  polynomial  of  degree  i  in  x^,  x^,  x^, 
the  number  of  constants  in  the  equation  can  be  readily  calculated. 
For  we  may  write 

n.  ==  (^o^V  +  <^ia-3'-^  H +  cf>i, 

4>k  being  a  homogeneous  polynomial  in  x^,  x^^  of  degree  A;  and  con- 
sequently containing  A;  +  1  constants.  The  number  of  constants 
in  XI-  is  therefore 

1  I  o  ,  ,  n  ;^^(^  +  l)(r  +  2)_(^  +  2)!. 

2  i!2! 

This  number  of  constants  in  u^  is  now  to  be  summed  for  all  inte- 
gral values  of  i  from  0  to  n.  By  induction  the  sum  is  readily 
found  to  be 

^      2^      3j_4  (n  +  l)(n+2)_(n  +  3)! 

22  2  n!3! 

which  is  the  number  of  homogeneous  coefficients  in  the  equation 
of  the  surface.     The  number  of  independent  conditions  which 
the  surface  can  satisfy  is  one  less  than  this  or 
(n  +  3) !  _  ^  ^  n^  +  6  w'  +  11  n 
w!3!  ~  6 

206 


Arts.  161-163]         INTERSECTION  WITH  A  LINE  207 

162.    Notation.     It  will  be  convenieut  to  introduce  the  follow- 
ing symbols  : 

A  f(x)  =  v  ^M  +  v  ^^  +  »  ^-f^  +  v  M^-         Vv  ^^ 


A//(x-)  =  V — — ■  yi^y-^yiUi 


d\f(x) 


wherein  1  ^  r  ^n  and  a,  b,  c,  d  are  positive  integers  (or  zero), 
satisfying  the  condition  a-f-6  +  c  +  d  =  r, 

EXERCISES  V 

Let  /(.x)  =  aioooXi*  +  ao4ooX2*  +  aooioX3*  +  aQoo4Xi*  +  6  02200X1^X2^+600220X2^X3^ 
+  6  ao202XrX4^  +  6  aoo22.'*;3-^4'^  +  G  a202o.i;i''^-^3"^  +  6  a2002Xi'Xi^. 

1.  Find  A//(x)  for  r  =  1,  2,  3,  4. 

2.  Show  that  A,  [A//(.r)]  =  A//(x). 

3.  Show  that  ^A,V(^)  =  Ax/(y). 

4.  Show  that  Ay-f(x)  =  Ax^fiy). 

5.  Show  that  A,f(x)  =4/(x)  ;    AxVC^:)  =  12/(x)  ;    A^VW  =  24/(x). 

163.  Intersection  of  a  line  and  a  surface.  If  (y),  (x)  are  any  two 
points  in  space,  the  coordinates  of  any  point  (z)  on  the  line  joining 
them  are  of  the  form  z^  =  \y^  +  fjiX^  (Art.  95).  If  (z)  lies  on 
f[x)  —  0,  then  f{\y  +  \x.x)  =  0.  By  Taylor's  theorem  for  the 
expansion  of  a  function  of  four  variables,  we  have,  since 
A/+*/(?/)  =  0  for  all  positive  integral  values  of  A;, 

Siky  +  /.X-)  =  A"/(^)  +  A-VA^/Cv)  +  - 

^A//(^)  +  -+-^ 
r!  n 


+  ^-^  A//(^)  +  • .  •  +  ;^  A/  fiy)  =  0.  (4) 


This  equation  may  also  be  written  in  the  form 

f{\y  +  iix)=  iL-fix)  +  /x"  iAA,/(.r)  + 


/• !  w : 


+  Lr^  A//(x)  +  -  +  ^  A,"/(x)  =  0,  (6) 


which  is  equivalent  to  the  preceding  one. 


208  CURVES   AND  SURFACES  [Chap.  XIII. 

If  these  equations  are  identically  satisfied,  the  line  joining  (y) 
to  (x)  lies  entirely  on  the  surface.  If  they  are  not  identically 
satisfied,  they  are  homogeneous,  of  degree  n  in  X,  fx  and  conse- 
quently determine  n  intersections  of  the  line  and  the  surface.  If 
we  define  the  order  of  a  surface  as  the  number  of  points  in  which 
it  is  intersected  by  a  line,  we  have  the  following  theorem. 

Theokem.  The  order  of  a  surface  is  the  degree  of  its  equation  in 
point  coordinates. 

164.  Polar  surfaces.  In  (5)  let  the  point  (y)  be  fixed  but  let 
(x)  vary  in  such  a  way  that  the  equation 

a;/(x)  =  o  (6) 

is  satisfied. 

This  equation  defines  a  surface  of  order  n  —  r  called  the  rth  polar 
surface  of  the  fixed  point  (y)  with  regard  to  the  given  surface 
/(.!•)  =  0.  When  r  =  n  —  l,  the  surface  (6)  is  a  plane.  It  is 
called  the  polar  plane  of  the  point  (y)  as  to  f(x)  =  0 ;  when 
r  =  n  —  2,  the  resulting  quadric  defined  by  (6)  is  called  the 
polar  quadric,  etc. 

In  the  identities  (4)  and  (5)  the  coeflicients  of  like  powers  of 
A,  fi  are  equal,  that  is, 

i-  a;  fix)  =      ^      A— /(y)- 

r  !  (n  —  r)  ! 

From  this  identity  we  have  the  following  theorem  : 

Theorem  I.  If  (x)  lies  on  the  rth  jjolar  of  (y),  then  (y)  lies  on 
the  (n  —  r)th  polar  of  (x). 

If  in  (4),  the  two  points  (y),  (x)  are  coincident,  then 
fi\x  +  ^)  =  (A  +  ^yf{x)  =  X-fix)  +  A-VA^/Cx)  +  •.. 

\n—r,,r 

By  expanding  (A  +  fi)"  by  the  binomial  theorem  and  equating 
coefficients  of  like  powers  of  A,  /a  in  the  preceding  identity,  we 
obtain 

A/f(x)=--^f(x), 
(w  —  ?•) ! 


Arts.  164,  165]  TANGENT   LINES  AND   PLANES  209 

which  is  called  the  generalized  Euler  theorem  for  homogeneous 
functions.     From  this  identity  we  hav^e  the  following  theorem  : 

Theorem  II.  The  locus  of  a  jwint  which  lies  on  any  one  and 
therefore  on  all  its  own  polar  surfaces  is  the  given  surface  f(x)  —  0. 

From  the  definition  of  ^Jf{x)  (Art.  162)  it  follows  that  if 

^  <'■<"'  A//(.r)  =  A/[A-/(aO]. 

From  this  identity  we  have  the  theorem  : 

Theorem  III.  Tlie  sth  polar  surface  of  the  (r  —  s)th  polar  siir- 
face  of  (y)  with  respect  to  f(x)  =  0  coincides  with  the  rth  polar  sur- 
face of  (y). 

EXERCISES 

1.  Determine  the  coordinates  of  tiie  points  in  which  the  line  Joining 
(1,  0,  0,  0)  to  (0,  0,  0,  1)  intersects  the  surface 

Xi'^  +  2  X^^  —  X3^  —  4  Xi^  +  4  Xi2X4  —  XiX4''^  +  6  X2'^X3  —  6  X1X2X.3  =  0. 

2.  Determine  a  so  that  two  intersections  of  the  line  joining  (0,  1,  0,  0) 
to  (0,  0,  1,  0)  with  the  surface 

Xl*  +  X2*  +  X-i*  +  Xi*  4-  0X2^X3  +  2  (a  -   1)X22X,32  +  4  X2X33  +  6  X1X2X3X4  =  0 

coincide. 

3.  Show  that  any  line  through  (1,  0,  0,  0)  has  two  of  its  intersections 
with  the  surface 

3  XzW  +  Xi*  +  6  Xi2X2-  +    12  X22X42  +  4  Xirs"^  +  24  Xi,r2X3X4  =  0 

coincident  at  (1,  0,  0,  0). 

4.  Prove  the  theorems  of  Art.  164  for  the  surface  of  Ex.  3  by  actual 
differentiation. 

165.  Tangent  lines  and  planes.  A  line  is  said  to  touch  a  sur- 
face at  a  point  P  on  it  if  two  of  its  intersections  with  the  surface 
coincide  at  P.  In  equation  (4)  let  (?/)  now  be  a  fixed  point  on 
the  given  surface  so  th.a.t  fy)  =  0.  One  root  of  (4)  is  now  /x  =  0, 
and  one  of  the  intersections  (x)  coincides  with  (y). 

The  condition  that  a  second  intersection  of  the  line  (y)(x)  coin- 
cides wdth  (y)  is  that  fj}  is  a  factor  of  (4),  that  is,  that  (x)  is  a 
point  in  the  plane 

dyi         5//2         a^/3         dy^ 

All  the  lines  which  touch  f(x)  =  0  at  (?/)  lie  in  the  plane  (7)  and 
every  line  through  (y)  in  this  plane  is  a  tangent  line.     This  plane 


210  CURVES   AND   SURFACES  [Chap.  XIII. 

is  called  the  tangent  plane  of  (>/).  The  plane  (7)  is  also  the  polar 
plane  of  (ij) ;  hence  we  have  the  theorem  : 

Theokkm.  The  polar  plane  of  a  point  P  on  the  surface  is  the 
tangent  plane  to  the  surface  at  P. 

From  Art.  164,  Theorem  III  it  also  follows  that  the  tangent 
plane  to  /(.r)  =  0  at  a  point  (y)  on  it  is  also  the  tangent  plane  at 
(y)  to  all  the  polar  surfaces  of  {y)  with  regard  tof(x)  =  0. 

166.  Inflexional  tangents.  A  line  is  said  to  have  contact  of 
the  second  order  with  a  surface  at  any  point  P  on  it  if  three  of 
its  intersections  with  the  surface  coincide  at  P. 

Let  (?/)  be  a  given  point  on  the  surface,  so  that/(?/)=0.  The 
condition  that  the  line  {y){z)  has  contact  of  the  second  order  at 
(y)  is  that  fx^  is  a  factor  of  (4),  that  is,  that  (2:)  lies  on  the  tangent 
plane  and  on  the  polar  quadric  of   (?/).     Hence  (2;)  lies  on  the 

intersection  of 

A,/(z/)=0,     A//(^/)  =  0. 

Since  A^/(?/)  =  0  is  the  tangent  plane  of  the  quadric  A//(y)=0 
at  (y),  the  conic  of  intersection  of  the  taiigent  plane  and  polar 
quadric  consists  of  two  lines,  each  of  which  has  contact  of  the 
second  order  with  _/(,r)  =0  at  the  point  (?/).  These  two  lines  are 
called  the  inflexional  tangents  to  the  surface  at  the  point  P.  The 
section  of  the  surface  by  an  arbitrary  plane  through  either  of 
these  lines  has  an  inflexion  at  (?/),  the  given  line  being  the  inflex- 
ional tangent. 

167.  Double  points.  A  point  P  is  said  to  be  a  double  point  or 
node  on  a  surface  if  every  line  through  the  point  has  two  inter- 
sections with  the  surface  coincident  at  P.  If  (y)  is  a  double 
point  on  f(x)  =  0,  equation  (4)  has  /x^  as  factor  for  all  positions 
of  (2),  that  is,  ^J{y)  =  0  is  an  identity  in  z^,  z.,,  z^,  z^.  It  follows 
that  if  (?/)  is  a  double  point,  its  coordinates  satisfy  the  four 
equations 

M2()=0,      M^  =  o,     M^  =  o,      M^=0.  (8) 

Conversely,  if  these  conditions  are  satisfied,  it  follows,  since 
nfijj)  =  ^yfQl),  that  equation  (4)  has  the  double  root  /x^  =  0  and 


Arts.  165-168]  FIRST  POLAR  SURFACE  211 

{y)  is  a  double  point.  Hence  the  necessary  and  sufficient  condi- 
tion that/(x')  =  0  has  a  double  point  at  {y)  is  that  the  coordinates 
of  {y)  satisfy  equations  (8).  Unless  the  contrary  assumption  is 
stated,  it  will  be  assumed  that/(ic)  =  0  has  no  double  points. 

EXERCISES 

1.  Impose  the  condition  that  the  point  (0,  0,  0,  1)  lies  on  the  surface 
f{x)  =  0  and  find  the  equation  of  the  tangent  plane  to  the  surface  at  that 
point. 

2.  Determine  the  condition  that  the  surface  of  Ex.  1  has  a  double  point 
at  (0,  0,  0,  1). 

3.  Show  that  the  point  (1,  1,  1,  1)  lies  on  the  surface  of  Ex.  1,  Art.  164, 
and  determine  the  equation  of  the  tangent  plane  at  that  point. 

4.  Find  the  equations  of  the  inflexional  tangents  of  the  surface  of  Ex.  1, 
Art.  164,  at  the  point  (1,  1,  1,  1). 

5.  Show  that  the  lines  through  a  double  point  on  a  surface  f{x)  =  0  which 
have  three  intersections  with  the  surface  coincident  at  the  double  point  form 
a  quadric  cone. 

6.  Show  that  there  are  six  lines  through  a  double  point  on  a  surface 
f(x)  =  0  which  have  four  points  of  intersection  with  the  surface  coincident 
at  the  double  point. 

7.  Prove  that  the  curve  of  section  of  a  surface  by  any  tangent  plane  has  a 
double  point  at  the  point  of  tangency,  and  the  inflexional  tangents  are  the 
tangents  at  the  double  point. 

168.  The  first  polar  surface  and  tangent  cone.  If  in  equation  (7), 
the  coordinates  Xi,  x.;,,  x.,  x^  are  regarded  as  fixed,  and  ?/i,  ?/2,  2/3,  y^ 
as  variable,  the  locus  of  the  equation  is  the  first  polar  of  the 
point  (x). 

Theorem  I.  Tlie  first  polar  surface  of  any  point  in  space  passes 
through  all  the  double  j^oints  of  the  given  surface. 

For,  if  f(x)=0  has  one  or  more  double  points,  the  coordinates 
of  each  must  satisfy  the  system  of  equations  (8)  and  also  (7). 

Theorem  II.  Tlie  joints  of  tangency  of  the  tangent  planes  to  the 
surface  from  a  point  {x)  lie  on  the  curve  of  intersection  of  the  given 
surface  and  the  first  polar  of{x). 

For,  if  (y)  is  the  point  of  tangency  of  a  tangent  plane  to 
f(x)=0  which  passes  through  the  given  point  (x),  the  coordi- 


212  CURVES  AND   SURFACES  [Chap.  XIII. 

nates  of  (y)  satisfy /(//)=  0  and  A^/'(^)  =  0,  Conversely,  if  (y) 
is  a  non-multiple  point  on  this  curve,  it  follows  that  the  tangent 
plane  at  {y)  passes  through  the  given  point  (ic). 

Since  the  line  joining  (x)  to  any  point  (y)  on  the  curve  defined 
in  Theorem  II  lies  in  the  tangent  plane  at  (y),  it  follows  that  it  is 
a  tangent  line.  The  locus  of  these  lines  is  a  cone  called  the 
tangent  cone  from  (x)  to  the  surface  f(x)  =  0.  To  obtain  the 
equation  of  this  cone  we  think  of  (x)  as  fixed  in  (4)  and  impose 
the  condition  on  (y)  that  two  of  the  roots  of  the  equation  in  A :  /x 
shall  be  coincident.     Hence  we  have  the  following  theorem : 

Theorem  III.  The  equation  of  the  tangent  cone  from  any  point 
(x)  is  obtained  by  equating  the  discriminant  of  (4)  to  zero. 

169.  Class  of  a  surface.  Equation  in  plane  coordinates.  A  point 
(x)  lies  on  tlie  surface /(a?)  =  0  if  its  coordinates  satisfy  the  equa- 
tion of  the  surface.  Similarly,  a  plane  (u)  touches  the  surface  if 
its  coordinates  satisfy  a  certain  relation,  called  the  equation  of 
the  surface  in  plane  coordinates.  The  class  of  a  surface  is  the 
dual  of  its  order ;  it  is  defined  as  the  number  of  tangent  planes  to 
the  surface  that  pass  through  an  arbitrary  line  and  is  equal  to 
the  degree  of  the  equation  of  the  surface  in  plane  coordinates. 

Theorem.  The  class  of  an  algebraic  surface  of  order  n,  having 
8  double  points  and  no  other  singularities,  is  n(7i  —  ly  —  2  8. 

Let  /(.^•)  =  0  be  of  order  n,  and  let  Pi=(y),  P.^={z)  be  two 
points  on  an  arbitrary  line  I.  The  point  of  tangency  of  every 
tangent  plane  to/(ic)  =  0  that  passes  through  I  lies  on  the  surface 
/(.c)  =  0,  on  the  polar  of  {y)  and  on  the  polar  of  (2;),  so  that  its 
coordinates  satisfy  the  equations 

/(.r)=0,     A,/(x)=0,     A,/(a;)  =  0. 

These  surfaces  are  of  orders  n,  n  —  1,  n  —  1,  respectively,  and 
have  n(n  —  1)^  points  in  common ;  \i  f{x)=:  0  has  no  double  points, 
each  of  these  points  is  a  point  of  tangency  of  a  plane  through  the 
line  I,  tangent  to  the  given  surface.  If /(a:)=0  has  a  double 
point,  Aj^/(x)=0  and  A,/(a7)  =  0,  both  pass  through  it,  hence  the 
number  of  remaining  sections  is  reduced  by  two. 

If  the  plane  u^x^^  +  u^x^  -{-  n^x^  +  u^x^  =  0  is  tangent  to  f{x)=  0 


Arts.  168-170]  THE   HESSIAN  213 

at  (.y),  then  this  plane  and  that  defined  by  equation  (7)  must  be 
identical,  hence 

Moreover,  (y)  lies  in  the  given  plane  and  also  on  the  given  sur- 
face,  hence      ^^^^^  _^  ^^^^^  ^  ^^^^  _^  ^^^^  ^  0^    ^^^^^  ^  0.  (10) 

If  between  (9)  and  (10)  the  coordinates  of  (y)  are  eliminated,  the 
resulting  equation  will  be  the  equation  of  the  given  surface  in 
plane  coordinates.  If  /(a;)  =  0  has  double  points,  the  resulting 
equation  will  be  composite  in  such  a  way  that  the  equation  of 
each  double  point  appears  as  a  double  factor. 

EXERCISES 

1.  Determine  the  equation  of  the  tangent  cone  to  the  surface 

Xi^  +  X2^  +  Xs^  +  Xi^  =  0 

from  the  point  (1,  0,  0,  0). 

2.  Write  the  equation  of  the  surface  of  Ex.  1  in  plane  coordinates. 

3.  Write  the  equation  of  the  surface 

a;ia:23;3  +  a:ia;3a;4  +  XiXoXt  +  XiX^Xt  =  0 
in  plane  coordinates. 

4.  Write  the  equation  of  the  surface  Xi'^x^  —  X2-X4  —  0  in  plane  coordinates. 

170.  The  Hessian.  The  locus  of  the  points  of  space  whose 
polar  quadrics  are  cones  is  called  the  Hessian  of  the  given  sur- 
face f{x)  =  0.  The  equation  of  the  polar  quadric  of  a  point  (x) 
may  be  written  in  the  form 

y^'my^y,  =  0,  (11) 

^  ax.  5.1V 

in  which  y^,  y^,  y^,  y^  are  the  current  coordinates.  This  quadric 
will  be  a  cone  if  its  discriminant  vanishes  (Art.  103),  hence  if  we 
put  for  brevity  ^  gy^^ 

ax-ax^ 
the  equation  of  the  Hessian  may  be  written  in  the  form 

/u  /12  /l3  J\i 

TT fl2  J 22  /23  X24 

./l3  /23  JZ3  /34 

7l4  ./24  /34  744 

It  is  of  order  4  (n  —  2). 


=  0.  (12) 


214  CURVES  AND   SURFACES  [Chap.  XIII. 

It  will  now  be  shown  that  the  Hessian  may  also  be  defined  as 
the  locus  of  double  points  on  first  polar  surfaces  of  the  given 
surface.     The  equation  of  the  first  polar  of  (y)  as  to/(a;)  =  0  is 

If  this  surface  has  a  double  point,  the  coordinates  of  the  double 
point  make  each  of  its  first  partial  derivatives  vanish,  by  (8),  thus 

ax^  axiox.^  dx^^dx^  dx^dXi 

yi^+2/.|^+ 2/3  ^  +  2/4^=0,  (13) 

0X16X2  OX^  0X20X3  0X20Xi 

Qlf  Q2f  Q2f  Q2f 

OXyOX:^  OX2OX3  0X3^  OX^OX^ 

oxiox^  dXiOX^  ox^ox^  oxv 

The  condition  that  these  equations  in  ^j,  ?/2>  2/3)  2/4  ^^^  consistent 
is  that  their  determinant  is  equal  to  zero,  that  is,  that  (x)  lies  on 
the  Hessian. 

171.  The  parabolic  curve.  The  curve  of  intersection  of  the 
given  surface  with  its  Hessian  is  called  the  parabolic  curve  on  the 
surface. 

Theorem.  At  any  point  of  the  parabolic  curve  the  two  inflexional 
tangents  to  the  surface  coincide. 

For,  let  (x)  be  a  point  on  the  parabolic  curve.  Since  (x)  lies  on 
the  Hessian,  its  polar  quadric  is  a  cone.  This  cone  passes  through 
(x)  (Art.  164).  The  inflexional  tangents  are  the  lines  which  the 
cone  has  in  common  with  its  tangent  plane  at  (x)  (Art.  166). 
These  lines  coincide  (Art.  121). 

172.  The  Steinerian.  It  was  just  seen  that  the  polar  quadric  of 
any  point  on  the  Hessian  is  a  cone.  Let  (.c)  be  a  point  on  H,  and 
(y)  the  vertex  of  its  polar  quadric  cone.  As  (x)  describes  H,  (y) 
also  describes  a  surface,  called  the  Steinerian  of  f(x)  =  0.  The 
polar  quadric  of  (;c)  is  given  by  equation  (11).  If  (y)  is  the  ver- 
tex of  the  cone,  its  coordinates  satisfy  (13).     The  equation  of  the 


Arts.  170-173]        ALGEBRAIC   SPACE   CURVES  215 

Steiuerian  jnay  be  obtained  by  eliminating  x^,  x^,  x^,  x^  from  these 
four  equations  (13).  As  the  equations  (13)  were  obtained  by  im- 
posing the  condition  that  the  first  polar  of  (?/)  has  a  double  point, 
we  may  also  define  the  Steinerian  as  the  locus  of  a  point  whose 
first  polar  surface  has  a  double  point  (lying  on  the  Hessian). 

EXERCISES 

1.  Prove  that  the  Hessian  and  the  Steinerian  of  a  cubic  surface  coincide. 

2.  Prove  that  if  a  point  of  the  Hessian  coincides  with  its  corresponding 
point  on  the  Steinerian,  it  is  a  double  point  of  the  given  surface,  and  con- 
versely. 

3.  Determine  the  equation  of  the  Hessian  of  the  surface 

ai-Ti^  +  a-ix-^  +  asa-s'  +  a^x.^  +  OsCx]  +  .ro  +  X3  +  X4)^  =  0. 

4.  Determine  the  order  of  the  Steinerian  of  a  general  surface  of  order  n. 

II.    Algebraic  Space  Curves 

173.  Systems  of  equations  defining  a  space  curve.  A  curve 
which  forms  the  complete  or  partial  intersection  of  two  algebraic 
surfaces  is  called  an  algebraic  curve ;  if  the  curve  does  not  lie  in 
a  plane,  it  is  called  a   space  curve. 

If  a  given  curve  C  forms  the  complete  intersection  of  two  sur- 
faces F^  =0,  i^2  =  ^1  so  that  the  points  of  C,  and  no  other  points, 
lie  on  both  surfaces,  then  the  equations  of  these  surfaces,  consid- 
ered as  simultaneous,  will  be  called  the  equations  of  the  given 
curve. 

If  the  intersection  of  the  surfaces  i^i  =  0  and  i^j  =  0  is  composite, 
and  G  is  one  component,  the  equations  2^^  =  0,  i^2  =  0  are  satisfied 
not  only  by  the  points  of  C  but  also  by  the  points  of  the  residual 
curve.  If  a  surface  i^3  =  0  through  G  can  be  found  which  has  no 
points  of  intersection  with  the  residual  curve  except  those  on  C, 
the  simultaneous  equations  F^  =  0,  i^2  =  0,  i^3  =  0  are  satisfied  only 
by  the  points  of  G  and  are  called  the  equations  of  the  curve. 

If  the  surfaces  F^  =  0,  i^^  =  0,  i^3  =  0  through  G  have  one  or 
more  points  in  common  which  do  not  lie  on  C,  then  a  fourth  sur- 
face i^j  =  0  can  be  found  through  G  which  does  not  contain  these 
residual  points,  but  may  intersect  the  residual  curve  of  J^i  =  0, 
7^2  =  0  in  other  points  not  on  2^3  =  0 ;  in  this  case  the  simultaneous 


216  CURVES  AND  SURFACES  [Chap.  XIII. 

equations  F,  =  0,  i^2  =  0,  i^s  =  0,  i^4  =  0  are  called  the  equations  of 
the  curve.  In  this  way  a  system  of  equations  can  be  found  which 
are  simultaneously  satisfied  by  points  of  C  and  by  no  others. 

As  an  illustration,  consider  the  composite  intersection  of  the 
quadric  surfaces 

It  consists  of  a  space  curve  and  the  line  X2  =  0,  x^  =  0.  The 
surface  XiX^  —  XzX^  =  0  also  contains  the  space  curve  since  it 
contains  every  point  common  to  the  quadrics  except  points 
on  the  line  X2  =  0,  ^3  =  0.  These  three  surfaces  are  sufficient 
to  define  the  curve.  The  surface  XiXi(xi  —  x^)—X2^-{-X2X^Xi=zO 
also  contains  the  given  curve.  It  does  not,  however,  with 
the  two  giveu  surfaces  constitute  a  system  whose  equations 
define  the  given  curve.  All  three  equations  are  satisfied,  not 
only  by  the  coordinates  of  the  points  of  the  curve,  but  by  the 
coordinates  of  the  point  (1,  0,  0,  1)  which  does  not  lie  on  the 
curve,  since  it  does  not  lie  on  the  surface  XiX^  —  x^x^  =  0.  The  sur- 
face x-^x^x^  +  x^  —  ;r./  —  xi  =  0  passes  througli  the  curve  but  not 
through  the  point  (1,  0,  0,  1).  The  curve  is  therefore  completely 
defined  by  regarding  the  four  equations 

XiXi  (Xi  —  x^)  —  xi  +  x^^Xi  =  0,         X1CC4  (a^i  +  X4)  —  xi  —  x^  =  0 
as  simultaneous. 

1 74.    Order  of  an  algebraic  curve.     Let  F^  =  0,  F^-  =  0  be  two 

surfaces  of  orders  /*,  /x',  respectively,  and  let  C  be  their  (proper 
or  composite)  curve  of  intersection.  Any  plane  that  does  not  con- 
tain C  (or  a  component  of  it)  intersects  C  in  fxfjj  points.  For, 
any  such  plane  intersects  F^  =  0  in  a  curve  of  order  /*,  and  inter- 
sects i^  -  in  a  curve  of  order  jx.  These  coplanar  curves  have 
precisely  jx^jJ  points  in  common.* 

It  can  in  fact  be  shown  that  every  algebraic  curve,  whether 
defined  as  the  complete  intersection  of  two  surfaces  or  not,  is 
intersected  by  any  two  planes,  neither  of  which  contains  the 
curve  or  a  component  of  it,  in  the  same  number  of  points.t     We 

*  See,  e.g.,  Fine:  College  Algebra  (1905),  p.  519. 

t  Halphen:  Jour,  de  I'ecole  polyteclinique.  Vol.  52  (1882),  p.  10. 


Arts.  173-175]  PROJECTING  CONES  217 

shall  assume,  without  proof,  the  truth  of  this  statement.  The 
number  of  points  in  which  an  arbitrary  plane  intersects  an  alge- 
braic curve  is  called  the  order  of  the  curve  (Art.  140). 

175.  Projecting  cones.  If  every  point  of  a  space  curve  is 
joined  by  a  line  to  a  fixed  point  P  in  space,  a  cone  is  defined, 
called  the  projecting  cone  of  the  curve  from  the  point  P.  If  the 
point  P  lies  at  infinity,  the  projecting  cone  from  P  is  a  cylinder 
(Art.  44).  Except  in  metrical  cases  to  be  discussed  later  we 
shall  make  no  distinction  between  cylinders  and  cones. 

For  an  arbitrary  point  P  an  arbitrary  generator  of  the  project- 
ing cone  intersects  the  curve  in  only  one  point.  It  may  happen, 
however,  for  particular  positions  of  the  point  P,  that  every 
generator  meets  the  curve  in  two  or  more  points.  If  in 
this  case  P  does  not  lie  on  the  curve  or  if  P  lies  on  the 
curve  and  every  generator  through  P  intersects  the  curve  in 
two  or  more  points  distinct  from  P,  the  curve  is  called  a  conical 
curve. 

Let  P  be  a  point  not  on  the  curve,  such  that  an  arbitrary 
generator  of  the  projecting  cone  from  P  meets  the  curve  in  just 
one  point.  The  order  of  the  projecting  cone  is  the  number  of 
generators  in  an  arbitrary  plane  through  its  vertex.  Each  gener- 
ator contains  one  point  on  the  curve,  hence  the  order  of  the  pro- 
jecting cone  is  equal  to  the  order  of  the  curve.  If  P  is  on  the 
curve,  the  order  of  the  projecting  cone  is  one  less  than  the  order 
of  the  curve. 

Theorem.  To  find  the  equation  of  the  2^'>'ojecting  cone  of  the 
simple  or  composite  curve  defined  by  the  complete  intersection  of  two 
surfaces,  from  a  vertex  of  the  tetrahedron  of  reference,  eliminate  be- 
tween the  equations  the  variable  which  does  not  vanish  at  that 
vertex. 

Let  the  equations  of  the  given  surfaces  be  P^  =  0  and  F^.  =  0 
and  let  it  be  required  to  project  the  curve  of  intersection  of  these 
surfaces  from  the  point  (0,  0,  0,  1). 

Let  (y)  be  any  point  of  space.  The  coordinates  of  any  point 
(x)  on  the  line  joining  (0,  0,  0,  1)  to  (y)  are  of  the  form 

X,  =  A?/i,     a'2  =  Xy2,     x^  =  Xy^,     x^  =  Xy^  +  o-. 


218  CURVES  AND  SURFACES  [Chap.  XIII. 

The  points  in  which  this  line  intersects  F^  =  0,  F^.  =  0  are  de- 
fined by 

F^,{x)  =  F^{Xy„  Xy„  Xy^,  Xy^  +  a) 

=  >^''Ffy„  y„  y„  y,+  f)  =  0, 

^  ^^  (14) 

F^.{x)  =  F^-{Xyi,  Xy^,  Xy^,  Xy^  +  a) 

=  Xt^'FJy„  y.„  2/3,  2/4+H=0, 

respectively.  The  condition  that  the  line  intersects  both  surfaces 
in  the  same  point  is  that  these  equations  have  a  common  root  in 

-,  hence  the  equation  of  the  projecting  cone  is  obtained  by  elimi- 
A 

nating  -  between  these  two  equations  (cf.  Art.  44).     If  -  is  elimi- 

A  A 

nated  from  (14),  y^  is  also  eliminated  and  the  resulting  surface  is 
identical  with  that  obtained  by  eliminating  x^  between  the  equa- 
tions of  the  given  surfaces. 

If  the  curve  of  intersection  is  composite,  the  projecting  cone  is 
composite,  one  component  belonging  to  each  component  curve. 

A  method  for  determining  the  projecting  cone  from  any  point 
P  in  space  may  be  deduced  by  similar  reasoning,  but  the  process 
is  not  quite  so  simple. 

EXERCISES 

1.  Show  that  the  intersection  of  the  surfaces 

3:1X2  —  X3X1  +  X^'^  —  X2X3  =  0,      XiXs'^  —  X1X2X4  +  X2(X42  —  X2X3)  =  0 

is  composite. 

2.  Represent  each  component  curve  of  Ex.  1  completely  by  two  or  more 
equations. 

3.  Find  the  equation  of  the  projecting  cone  of  the  curve 

Xi'^  +  X32  +  X42  +  2  X1X4  =  0,     X42  +  2  X2X4  -  Xi^  +  2  X32  =  0 
from  the  point  (0,  0,  0,  1). 

4.  Find  the  equation  of  the  projecting  cone  of  the  curve 

Xi2  +  4  X32  -  X42  =  0,     Xi2  _  2  X22  +  2  X32  -  3  .r4-  =  0 
from  the  point  (0,  0,  0,  1). 

5.  Find  the  equation  of  the  projecting  cone  of  the  curve 

xi"^  +  X2'^  +  X32  4-  xt^  =  0,     aixi2  +  a2X:2  -f-  03X32  +  a^Xi^  =  0 
from  the  point  (0,  0,  0,  1), 


Arts.  175,  176]  MONOIDAL  REPRESENTATION  219 

6.  Show  by  means  of  elimination  that,  if  (0,  0,  0,  1)  does  not  lie  on  the 
curve  Ffj^=  0,  i^^-^O,  the  order  of  the  projecting  cone  from  (0,  0,  0,  1)  is  ix/x', 
provided  the  curve  is  not  conical  from  (0,  0,  0,  1). 

7.  Find  the  equation  of  the  projecting  cone  of  the  curve 

Xi^  +  2  X2^  —  Xs^  —  0,     Xi'  —  0*2^:3  +  Xi^  =  0 
from  the  point  (1,  1,  1,  1). 

176.  Monoidal  representation.  If  a  non-composite  space  curve 
(7„  of  order  m  is  defined  as  the  complete  or  partial  intersection 
of  two  surfaces  F^  =  0,  F^.  =  0,  other  surfaces  on  which  the  curve 
lies  can  be  obtained  from  the  given  ones  by  algebraic  processes. 
Among  such  surfaces  we  have  already  discussed  the  projecting 
cone  from  a  given  point  P.  We  shall  now  show  how  to  obtain 
the  equation  of  a  surface  which  contains  C„  and  has  at  P  a  point 
of  multiplicity  one  less  than  the  order  of  the  surface.  Such  a 
surface  is  called  a  monoid. 

In  determining  the  equation  of  a  monoid  through  C„,  we  shall 
assume  that  neither  the  complete  intersection  of  P^  =  0  and 
F^.  =  0  nor  any  component  of  it  is  a  conical  curve  from  P.  We 
shall  also  assume  that  P  does  not  lie  on  this  curve  of  intersection. 

Let  P  be  chosen  as  (0,  0,  0,  1)  and  let  the  equations  F^  =  0, 
F^.  =  0  be  arranged  in  powers  of  x^  (Art.  161). 

F^  =  u^,^  +  Kior/"'  +  •••  +  w^  =  0, 
F^-  =  iVV'  +  ^iaJ4''"'  +  •••  +v^-  =  0, 

wherein  ?^,-,  v,-  are  homogeneous  functions  of  x^,  x^,  x-^  of  degree  i. 
Let  the  notation  be  so  chosen  that  /x'  ^  fx.     The  equation 

v^/->'F^  -  u,F^.  =  0 

contains  x^  to  at  most  the  power  /a'  —  1.     The  surface  represented 
by  it  passes  through  the  curve  C„,  since  the  equation  is  satisfied 
by  the  coordinates  of  every  point  which  satisfy  F^  —  0  and  F^.  =  0. 
The  equation 

V  .F  -u  F  .  =  0 

MM  MM 

is  divisible  by  x^.  If  this  factor  is  removed,  the  resulting  equation 
is  of  degree  at  most  /x'  —  1  in  .^4,  and  determines  a  surface  which 
passes  through  C'„. 

If  either  of  these  equations  contains  x^  to  the  first  but  to  no 


220  CURVES  AND   SURFACES  [Chap.  XIII. 

higher  power,  the  surface  determined  by  it  is  of  the  type  required. 
If  not,  the  two  equations  cannot  both  be  independent  of  x^  nor 
can  they  coincide,  since  in  that  case  the  curve  F^  =  0,  F^-  =  0 
would  be  conical  from  (0,  0,  0,  1). 

By  applying  this  same  process  to  the  two  equations  just  ob- 
tained, we  may  obtain  two  new  ones  which  contain  x^  to  at  most 
the  power  fi  —  2. 

Continuing  in  this  way  with  successive  partial  elimination,  we 
obtain  finally  an  equation  of  the  form 

M=  x^(f}„_y{Xi,  X2,  Xs)  -  <^„(a^i,  X2,  Xs)  =  0, 

in  which  <^„_i  and  (^„  are  homogeneous  functions,  not  identically 
zero,  of  degree  n  —  1  andn,  respectively,  in  x^,  X2,  Xy  The  surface 
M  =  0  is  of  order  n  and  has  an  (it  —  l)-fold  point  at  (0,  0,  0, 1).  It 
is  consequently  a  monoid.  The  surface  <^„  =  0  is  a  cone ;  it  is 
called  the  superior  cone  of  the  monoid.  If  ?i  >  1,  <^„_i  =  0  is  the 
equation  of  another  cone,  called  the  inferior  cone  of  the  monoid. 

Let/„(.Ti,  X2,  .T3)  =  0  be  the  equation  of  the  projecting  cone  from 
(0,0,0,1).     The  equations 

are  said  to  constitute  a  monoidal  representation  of  the  curve  C„. 
The  advantage  of  this  representation  is  that  the  residual  inter- 
section, if  any,  of  the  two  surfaces  M  =  0,  f„  =  0  consists  of  lines 
common  to  the  three  cones 

/„  =  0,     <^„,  =0,     0„  =  O. 

For,  let  P  be  a  point  common  to  /„  =  0,  M—  0,  but  not  lying  on 
C„,  nor  at  the  vertex  (0,  0,  0,  1).  The  generator  of /„  =  0  passing 
through  P  intersects  C„  in  some  point  P'.  Since  this  generator 
has  P,  P'  and  n  —  1  points  at  (0,  0,  0,  1)  in  common  with  M  =  0, 
it  lies  entirely  on  the  monoid  (Art.  163).  For  every  point  of  this 
line,  that  is,  independently  of  the  value  of  x^,  the  equation 
Xi(f)„_i  —  cf)^  =  0  must  be  satisfied ;  hence  the  given  generator  lies 
on  the  inferior  cone  and  on  the  superior  cone. 

It  follows  at  once  from  the  above  discussion  that  if  any  genera- 
tor of  /„  =  0  intersects  C„  in  two  points  P,  Q,  it  lies  entirely  on 
the  monoid  and  forms  a  part  of  the  residual  intersection.  Such  a 
line  is  called  a  double  generator  of  the  projecting  cone,  since,  in 


Arts.  176,  177]       NUMBER  OF   INTERSECTIONS  221 

tracing  the  curve,  the  generator  takes  the  position  determined  by 
Pon  C„  and  also  the  position,  coincident  with  the  first,  determined 
by  Q.  Every  such  line  counts  for  two  intersections  of  M=  0 
and  .4  =  0. 

Each  of  these  bisecants  of  the  curve  is  said  to  determine  an 
apparent  double  point  of  C„  from  (0,  0,  0,  1) ;  the  curve  appears 
from  (0,  0,  0,  1)  to  have  a  double  point  on  each  of  these  lines. 

It  can  be  proved*  and  will  here  be  assumed  that  the  number 
of  apparent  double  points  of  a  given  curve  is  the  same  for  every 
point  not  lying  on  it,  except  the  vertices  of  the  cones,  if  any,  on 
which  Cn  is  a  conical  curve.     This  number  will  be  denoted  by  h. 

We  shall  now  show  that  if  a  point  P  which  does  not  lie  on  C„, 
nor  on  any  line  that  intersects  C„  in  more  than  two  points,  nor  at 
the  vertex  of  a  cone  (if  any)  of  bisecants  to  C„„  is  chosen  for  the 
vertex,  then  the  order  of  the  monoid  from  P  is  at  least  half  the 
order  of  C„. 

The  complete  intersection  of  the  projecting  cone/„  =  0  and  the 
monoid  a;4<^„  i  —  </>„  =  0  is  a  curve  of  order  mn.  The  curve  C^  is 
one  component  of  order  m,  and  the  h  bisecants  of  C^  through 
(0,  0,  0,  1)  together  form  a  component  of  order  2h.  If  the  num- 
ber of  residual  intersecting  lines  is  denoted  by  k,  then 

mn  —  m  —  2  h  =  Jc,    k  ^  0. 

The  h  bisecants  of  C„  and  the  k  residual  lines  are  components  of 
the  intersection  of  <^„_i  =0,  <^„  =  0.     Hence 


from  which 
and 


7i[u  —  l)  =  h-}-k  =7n()i  —  1)—  h, 
{m  —  n){n  —  l)  =  h^  -  (n  —  1), 
>  7n 


n 

—  v  ' 

which  proves  the  proposition. 

177.   Number  of  intersections  of  algebraic  curves  and  surfaces. 

Theorem.     Any  surface  of  order  jj.  lohich  does  not  contain  a  given 
non-composite  curve  of  order  m  intersects  it  in  mfx  j)oints. 

*N()etlier:  Ziir  Grumllegiing  der  Tlv^orie  der  alfjebraischeii  Raumkurven,  Ab- 
handlungen  der  k.  preussisclien  Akademie  der  Wissensehaften  fiir  1S82. 


222  CURVES   AND   SURFACES  [Chap.  XIII. 

Let  C„  be  the  given  curve  and  i^^^  =  0  be  the  equation  of  the 
given  surface.  Choose  (0,  0,  0,  1)  not  on  F^  =  0,  and  let  the 
monoidal  equations  of  C„  be  /„  =  0,  Xi<f>^_i  —  <^^  =  0.  The  com- 
plete intersection  of /„  =  0,  .t^4<^„_i  —  <^„  =  0  consists  of  (7„  and  of 
m{n  —  l)  lines  through  (0,  0,  0,  1).  As  F^=0  does  not  pass 
through  (0,  0,  0,  1),  it  cannot  contain  any  of  these  lines.  Hence 
Ffj,  =  0,  /„  =  0,  M^  =  0  have  no  common  component.  They  con- 
sequently intersect  in  mri/j.  points.  Of  these,  mfx{n  —  1)  points  are 
where  the  residual  lines  intersect  F^=  0.  The  remaining  m/x  points 
lie  on  C„.  If  C„  has  vi/x.  +  1  points  on  F^  =  0,  it  lies  on  the  sur- 
face, for  the  three  surfaces  /^  =  0,  M^  =  0,  F^  =  0  have  now 
mnfx  +  1  points  in  common,  and  therefore  all  contain  a  common 
curve.  Since  the  lines  do  not  lie  on  ^^^  =  0,  and/„  =  0,  3/"^  =  0 
have  no  other  component  curve  except  C„,  it  follows  that  C„  must 

lie  on  F^  =  0. 

EXERCISES 

1.  Show  that  a  plane  or  any  proper  quadric  is  a  monoid. 

2.  Write  the  equation  of  a  monoid  of  order  three. 

3.  Show  that  the  only  curve  of  order  one  is  a  line. 

4.  Show  that  the  only  irreducible  curve  of  order  two  is  a  conic. 

5.  Show  that  a  composite  curve  of  order  two  exists  which  does  not  lie  in 
a  plane.     How  many  apparent  double  points  has  this  curve  ? 

6.  Show  that  a  bundle  of  quadrics  pass  through  a  proper  space  cubic  curve. 

7.  Write  a  monoidal  representation  of  a  space  cubic  curve. 

8.  Show  that  every  irreducible  curve  of  order  four  lies  on  a  quadric 
surface. 

9.  Discuss  the  statements  of  Exs.  6  and  8  for  the  case  of  composite  cubics 
and  composite  quartics. 

178.  Parametric  equations  of  rational  curves.  Since  a  space 
curve  is  defined  as  the  complete  or  partial  intersection  of  two 
surfaces,  the  coordinates  of  its  points  are  functions  of  a  single 
variable.  The  expressions  for  the  coordinates  of  a  point  as  func- 
tions of  a  single  variable  may  not  be  rational.  A  curve  which 
possesses  the  property  that  all  its  coordinates  can  be  expressed 
as  rational  functions  of  a  single  variable  is  called  a  rational  curve. 
"By  definition  the  equations  of  such  a  curve  can  be  written  para- 
metrically  in  the  form 

^i  =.W)  =  tt.o^"'  +  (I J""''  +  -  +  «•>.,    i  =  1,  2,  3,  4. 


Arts.  177,  178]  PARAMETRIC  EQUATIONS  223 

Since  the  variables  x^  are  homogeneous,  it  is  no  restriction  to 
suppose  that  the  polynomials  /^O  liave  no  common  factor.  To 
every  value  of  t  corresponds  a  unique  point  {x)  on  the  curve,  but 
it  may  happen  that  more  than  one  value  of  t  will  define  the  same 
point  (x)  on  the  curve.  If,  for  example,  the  functions  fi{t)  can 
be  expressed  in  the  form 

in  which  F^  are  homogeneous  rational  functions,  of  the  same  order, 
of  the  two  polynomials  <j>{t),  i}/{t),  then  f-(t)  will  define  the  same 
point  for  every  value  of  t  that  satisfies  the  equation 

where  s  is  given.  In  this  case  the  coordinates  of  the  points  on 
the  curve  are  rational  functions  of  s. 

Conversely,  it  will  now  be  shown  that  if  to  each  point  (x)  of  the  curve 
correspond  7i  values  of  t{n^  1),  then  t  may  be  replaced  by  a  new  variable, 
in  terms  of  which  the  correspondence  between  it  and  the  point  (x)  on  the 
curve  is  one  to  one. 

Let  fi,  t2,  •••,  tn  all  correspond  to  the  same  point  (x).     The  expressions 

Mt)Mh)-Mti)Mt)       i,  k  -  1,  2,  3,  4 

vanish  for  t  =  ti,  tz,  •••,  t„,  hence  they  have  a  common  factor  of  order  n, 
whose  coefficients  contain  ti, 

<t>o{h)t^  +  0i(«i)«"-i  +  •••  +  <t>n{h)- 

If  ti  is  replaced  by  ti,  the  expression  must  have  the  same  factor,  hence  the 
function 

<po{t2)t^  +  <Pl{t-l)t''-'^  +  •••   +  <Pn{t2) 

has  the  same  roots.  Similarly  for  tz,  •••,  «„■  It  follows  that  the  ratios  of  the 
coefficients 

00  :  01  :  •••  0„ 

have  the  same  values  for  ti,  h,  •••,  <„.  These  ratios  cannot  be  constant  for 
every  point  (x)  on  the  curve,  since  in  that  case  ti,  ■••,  ?„  would  be  independent 
of  (x),  contrary  to  hypothesis.     If  we  now  put 

and  eliminate  t  between  this  equation  and  x^  =fi{t),  the  resulting  equations 
may  be  written  in  the  form 

X.-  =  bioSP  +  biisp-'^  +  ■•■  +  ft.p, 
in  which  np  —  m. 


224  CURVES  AND  SURFACES  [Chap.  XIII. 

When  the  correspondence  between  (x)  and  t  is  one  to  one,  the 
order  of  the  curve  x^  =fi(t)  is  m.  For,  to  each  point  of  intersection 
of  the  curve  with  an  arbitrary  plane  '^u-x^  =  0  corresponds  a  root 
of  the  equation  '^Uifi{t)  =  0,  and  conversely. 

179,  Tangent  lines  and  developable  surface  of  a  curve.  Let  C  be 
the  intersection  of  two  algebraic  surfaces  F=0,  F'  =0  and  let  P 
be  an  arbitrary  point  on  C.  The  line  t  of  intersection  of  the 
tangent  planes  to  i^=  0  and  i^'  =  0  at  P  has  two  points  in  common 
with  each  of  the  surfaces  coincident  at  P  (Art.  165),  and  hence 
witli  C.  The  line  is  called  the  tangent  line  to  the  curve  C  at  the 
point  P.  The  locus  of  the  tangent  lines  to  C  is  a  ruled  surface. 
This  surface  is  called  the  developable  surface  of  C.  Its  equation 
may  be  found  by  eliminating  the  coordinates  y^,  y^,  ys,  2/4  of  -f* 
between  the  equations  of  O  and  of  the  tangent  planes,  thus : 

F(y)  =  0,  F'(y)  =0,X  -.  "f^  =  0,  2  -.■  ^^  =  0- 
^        dyi  ^         dyi 

Example.     The  intersection  of  the  surfaces 

xi^  +  X22+  xs^  +  Xi^  =  0,  aixi2  +  02052'^  +  03X32  +  04X42  =  0 

is  a  quartic  curve.     The  equation  of  the  developable  surface  of  this  quartic 
is  obtained  by  eliminating  yi,  2/2,  Vs,  Vi  between  the  equations 

yi^  +  2/2^  +  2/3^  +  ^4^  =  0,    ai2/l2  +  022/2^  +  032/3^  +  042/4=^  =  0, 

X12/1  +  X22/2  +  X32/3  +  X42/4  =  0, 
aiXiyi  +  a2X22/2  +  03X32/3  +  04X42/4  =  0. 

If  we  write  Uik  for  a,-  —  o^,  the  result  may  be  written  in  the  form 

4  ai2ai3a42a4:?(oi3Xi2  +  a23a;2'^  +  043X42)  (a-nxr  +  023X3^  +  a34X42)x22x42 

—  [0230l4'-Xi2x4-^  +  aoiQirXi^Xi^  +  03401 22Xi'-X22  +  a-iiau^Xs^Xi^  +  03l024'^X22X42 
+  003(012034 +  Ol3O24)X2'-X32]-  =  0. 

The  number  of  tangents  to  the  curve  C„  which  meet  an 
arbitrary  line  is  called  the  rank  of  the  curve.  From  this  defini- 
tion it  follows  that  the  rank  is  equal  to  the  order  of  the  develop- 
able surface.  It  is  the  same  number  for  every  line  not  on  the 
surface  (Art.  163). 

180.  Osculating  planes.  Equation  of  a  curve  in  plane  coordi- 
nates. Every  plane  through  the  tangent  line  to  C  at  P  contains 
the  line  and  has  therefore  two  points  in  common  with  G  at  P. 


Arts.  179,  180]  OSCULATING  PLANES  225 

Such  a  plane  is  called  a  tangent  plane.  Among  the  tangent  planes 
there  is  one  having  three  intersections  with  C  at  P.  This  plane 
is  called  the  osculating  plane  to  G  at  F.  The  number  of  osculating 
planes  to  C  which  pass  through  an  arbitrary  point  in  space  is 
called  the  class  of  C.  This  number  is  the  same  for  every  point  in 
space.*  If  G  is  the  intersection  oi  F=0  and  F'  -—^,  we  can 
obtain  two  equations  which  m.ust  be  satisfied  by  the  coordinates 
of  the  osculating  planes  of  G  by  eliminating  two  of  the  variables, 
as  .^3,  x^,  between  the  equations  F  =0,  F'  =  0,  and  the  equation  of 
the  plane  S^^a:-  =  0,  then  imposing  the  condition  that  the  resulting 
homogeneous  equation  in  the  other  two  variables  has  a  triple  root. 

Example.  The  two  surfaces  Xi'^  +  2  X2.r4  =  0,  Xo- +  2  a^iXs  =  0  intersect 
in  the  line  xi  =  0,  a;2  =  0  and  a  space  cubic  curve.  If  between  the  first  equa- 
tion and  SMjXj  —  0  we  eliminate  Xi,  we  find 

UiXi^  —  2  ?<iXiX2  —  2  M23C2^  —  2  U3X2X3  =  0. 

Now  if  we  eliminate  X3  between  this  result  and  the  second  given  equation, 
we  obtain 

M4X13  —  2  U1X1-X2  —  2  jtoXi.Ci^  +  Usx^^  =  0. 

Finally,  if  this  cubic  has  three  equal  roots,  its  first  member  must  be  a  cube. 
Hence 

2  ?<i2  +  .3  ?«4W2  =  0,    2  ?<22  -f-  3  UiUs  =  0. 

A  system  of  two  or  more  equations  in  plane  coordinates  (Art.  173) 
which  are  satisfied  by  the  coordinates  of  the  osculating  planes  of 
G,  and  by  no  others,  is  said  to  define  the  curve  C  in  plane  coor- 
dinates. To  a  curve  G  defined  in  this  way  may  be  applied  a  dis- 
cussion dual  to  that  given  in  Arts.  174-179. 


EXERCISES 

1.  Find  a  system  of  parametric  equations  of  the  rational  curve 

XlXo  —  X3X4  =  0,     X2X3  =  Xl'^  —  X'r^. 

2.  Write  the  equation  of  the  developable  surface  of  the  cubic  curve  lying 
on  the  surfaces 

xi-  -4-  2  X2X4  =  0,  xi^  +  2  X1X3  =:  0. 

3.  Find  two  equations  satisfied  by  the  cooi-dinates  of  the  osculating  planes 
of  the  curve 

X1X2  —  X3X4  =  0,    X2-  =  X32  +  X42, 

*  See  reference  in  Art.  176. 


226  CURVES  AND  SURFACES  [Chap.  XIII. 

4.  Define  the  dual  of  the  projecting  cone  of  a  curve  and  show  how  its 
equation  may  be  obtained, 

5.  Derive  the  dual  of  a  monoid al  representation  of  a  curve. 

6.  Define  the  dual  of  an  apparent  double  point. 

7.  What  is  the  dual  of  the  rank  of  a  space  curve  ? 

181.  Singular  points,  lines,  and  planes.  A  point  P  on  a  curve 
is  called  an  actual  double  point  if  two  of  the  points  of  intersection 
of  C  with  any  plane  through  P  coincide  at  P.  If  the  two  tangent 
lines  to  (7  at  P  are  distinct,  the  point  is  called  a  node.  If  the  two 
tangents  at  P  coincide,  the  point  is  called  a  cusp  or  stationary 
point.  Curves  may  have  higher  point  singularities,  for  example, 
a  curve  may  pass  through  the  same  point  three  or  more  times,  etc., 
but  such  singularities  will  not  be  considered  here. 

A  plane  is  said  to  be  a  double  osculating  plane  if  it  is  the  oscu- 
lating plane  at  two  points  on  the  curve.  A  plane  having  four 
points  of  intersection  with  the  curve  coincident  at  P  is  called  a 
stationary  plane. 

A  line  is  called  a  double  tangent  if  it  touches  the  curve  in  two 
distinct  points.  If  a  tangent  line  has  three  coincident  points  in 
common  with  the  curve,  it  is  called  a  stationary  or  an  inflexional 
tangent.     The  point  of  contact  is  called  a  linear  inflexion. 

182.  The  Cayley-Salmon  formulas.  We  shall  now  collect,  for 
the  purpose  of  pointing  out  certain  relations  existing  among  them, 
the  following  numbers  associated  with  a  given  space  curve.  We 
shall  assume  that  these  numbers  are  fixed  when  the  curve  is  given, 
and  are  independent  of  the  arbitrarily  chosen  plane,  line,  or  point 
that  may  be  used  to  determine  them. 

Given  a  space  curve  C.     Let 
m  =  its  order  (Art.  140). 
71  =  its  class  (Art.  180). 
r  =  its  rank  (Art.  179). 
^  =  the  number  of  its  nodes  (Art.  181). 
h  =  the  number  of  its  apparent  double  points  (Art.  176). 
g  —  the  number  of  lines  of  intersection  of  two  of  its  osculating 

planes  which  lie  in  a  given  plane  (dual  of  h). 
G  =  number  of  double  osculating  planes  (Art.  181). 
a  =  the  number  of  its  stationary  planes  (Art.  181). 


Arts    181,  182]       THE   CAYLEY-SALMON  FORMULAS       227 

(3  =the  number  of  its  stationary  points  (Art.  181). 

V  =  the  number  of  its  linear  inflexions  (Art.  181). 

(0  =  the  number  of  its  actual  double  tangents  (Art.  181). 

X  =  the  number  of   points  lying   in   a   given   plane,    through 

which  pass  two  distinct  tangents  to  C. 
y  =  the  number  of  planes  passing  through  a  given  point,  which 

contain  two  distinct  tangents  to  C. 

These  numbers  are  connected  by  certain  equations  called  the 
Cayley-Salmon  formulas ;  they  are  derived  from  the  analogous 
equations,  known  as  Plucker's  formulas,  connecting  the  character- 
istic numbers  of  plane  curves.  Let  fi  =  order,  v  =  class,  8  =  num- 
ber of  double  points,  t  =  number  of  double  tangents,  k  =  number 
of  cusps,  I  =  number  of  inflexions,  of  an  algebraic  plane  curve. 
Plucker's  formulas  are  * 

v  =  /t(/x— 1)  —  28  —  3  k;      i=3/A(/i,  —  2)  —  68  —  8k; 
/u.  =  v(v  —  1)  —  2t  —  3i;      K=3v(v  —  2)  —  6t  —  8 1. 

Those  in  the  second  line  are  the  duals  in  the  plane  of  those  in  the 
first  line. 

Let  the  given  space  curve  C  be  projected,  from  an  arbitrary 
point  P  not  lying  on  it,  upon  an  arbitrary  plane  not  passing 
through  P.  The  plane  curve  then  has  the  following  characteristic 
numbers : 

/x  =  m,  v=r,  8  =  ^  +  i?,  T=2/-t-w,  K  =  /8,  t  =  n-}-v. 

By  substituting  in  the  Pllicker  formulas,  we  obtain 

r  =  m(m-l)-2(h  +  H)-3(3; 
n+v  =  3m{m-2)-6{H+h)-Sfi;  . 

m  =  r(r—l)  —  2(y  +  oi)—3{n  +  v);  ^    ' 

;8  =  3  r(r  -  2)  -  6(2/  +  to)  -  8 (?i  +  ^). 

By  duality  in  space,  that  is,  by  taking  the  section  of  the  develop- 
able surface  by  an  arbitrary  plane,  we  have 

r  =  n  (u  -  1)  -  2  (G  +  ^)  -  3  « ; 
m-f-^  =  3n(n-2)-6(6?  +  gr)-8«; 

w  =  r(r-l)-2(a;-|-a)) -3(m+v);  ^^ 

a  =  3  /•(/•  -  2)  -6(x  -t-  oi)  -  8 (»i  +  V). 

♦Salmon:  Higher  Plane  Curves,  3d  edition  (1879).     See  p.  66. 


228  CURVES  AND   SURFACES  [Chap.  XIII. 

Of  these  eight  equations,  six  are  independent.  One  relation 
exists  among  the  first  set  of  four,  and  one  relation  among  the 
second  set. 

The  genus  of  a  curve  is  the  difference  between  the  sum  of  its 
apparent  and  actual  double  and  stationary  points  and  the  maxi- 
mum number  of  double  points  which  a  non-composite  plane  curve 
of  the  same  order  may  have.  If  the  genus  of  the  space  curve  C 
is  denoted  by  p,  we  have 


i> 


^(m-l)(m-2)_^^_^^^^^^(n-l)(n-2)_^^^^^^^^ 


183.  Curves  on  non-singular  quadric  surfaces.  It  has  been  seen 
(Art.  115)  that  the  equation  of  any  non-singular  quadric  surface 
may  be  reduced  to  the  form 

^'l^2  —  •''^3^4  =  0>  (1^) 

and  that  through  each  point  of  the  surface  passes  a  generator  of 
each  regulus  of  the  two  systems 

x^  —  Xx^  =  0,     a'a  —  Xx2  =  0,  (18) 

a-3  —  ixx^  =  0,     X2  —  fxx^  =  0.  (18') 

The  coordinates  of  the  point  of  intersection  of  the  generator 
A=  constant  with  the  generator  /ia  =  constant  are  (Art.  115) 

pXi  ^  A,     px^  =  [X,.    px^  =  Xjx,     px^  =  1.  (19) 

Consider  the  locus  of  the  points  whose  parameters  A,  p.  satisfy  a 
given  equation  /(A,  /x)  =  0,  algebraic,  and  of  degree  A;,  in  A  and  of 
degree  Jc2  in  p..  The  curve  /(A,  p.)=0  meets  an  arbitrary  generator 
p.  =  constant  in  ki  points,  and  an  arbitrary  generator  A  =  constant 
in  k^  points.  It  will  be  designated  by  the  symbol  [Aij,  k2^.  The 
order  of  the  curve  is  k^  4-  A^j,  since  the  plane  determined  by  any  two 
generators  of  different  reguli  meets  the  curve  in  k^  +  Atj  points  on 
these  two  lines,  and  nowhere  else. 

By  replacing  A,  p.  in  /(A,  p.)  =  0  by  their  values,  we  see  from  (17), 
(18),  (18')  that  the  curve  is  the  intersection  of  the  two  surfaces 


\Xi    xj 


Arts.  182,  183]       CURVES  ON  QUADRIC  SURFACES  229 

The  second  is  a  monoid  of  order  two  (Art.  176)  and  the  first  is  a 
cone  with  vertex  at  (0,0,  1,  0),  a(2  — l)-fold  point  on  the  monoid. 
Thus,  these  equations  constitute  a  particular  monoidal  representar 
tion  of  the  curve.  The  equations  of  the  image  (Art.  118)  of  the 
given  curve  on  the  plane  Za  =  0  are 


/A^^^O,   .^3  =  0. 
Va-4  xj 


The  two  generators  to  the  quadric  through  the  vertex  of  the  cone 
/=0  meet  the  plane  in  the  points  (1,  0,  0,  0),  (0,  1,  0,  0).  The 
former  is  a  Avfold  point  on  the  plane  curve,  and  the  latter  a 
^■l-fold  point. 

Theorem  I.  Tvoo  curves  of  symbols  [k^^,  k^'],  [k\,  A-'.,]  on  the  same 
non-singular  quadric  intersect  i)i  kjc'.,  +  k2k\  x^oints. 

Let  C,  C"  be  the  given  curves  of  symbols  [A^i,  A-.,],  [k\,  A'2],  re- 
spectively, and  let  the  equation  of  the  quadric  be  reduced  to  the 
form  (17)  in  such  a  way  that  the  point  (0,  0, 1,  0)  does  not  lie  on 
either  curve,  and  that  the  generators  .Ti  =  0,  x^  =  i)\  .1-2  =  0,  x^  —  i) 
through  (0,  0,  1,  0)  do  not  pass  through  a  point  of  intersection  of 
the  given  curves.  Project  the  curves  from  (0,  0,  1,  0).  Their 
images  on  x^  =  0  are  of  orders  k^  +  k^,  k\  +  k\,  respectively ;  they 
intersect  in  {k^  +  k^{k\  +  A'j)  points.  Of  these  points,  kji\  coin- 
cide at  (0,  1,  0,  0)  and  kjk'^^  at  (1,  0,  0,  0).  They  are  the  projections 
of  the  points  in  which  the  curves  meet  the  generators  passing 
through  (0,  0,  1,  0),  the  vertex  of  the  projecting  cone,  and  are 
therefore  apparent,  not  actual,  intersections  of  the  space  curves. 
The  remaining 

(ki  ~p  koj(k  I  -\-  tC  2)  —  iCifC  J  —  rCoK  2  ^^  fC^fC  2  ~r"  '12"'  1 

intersections  of  the  plane  curves  are  projections  of  tlie  actual  in- 
tersections of  the  space  curves,  hence  the  theorem  is  pioved. 

Theorem  II.  Tlie  number  of  ap^jarent  double  points  of  a  carve 
of  symbol  [Aj,  ^2]  on  a  quadric  is 

h  =  ^{k,^  +  k2'-k,-k2). 

Through  an  arbitrary  point  0  on  the  surface  pass  only  two  lines 
which  meet  the  curve  in  more  than  one  point,  namely,  the  two 
generators    passing    through    ().      The    generator    fx  =  constant 


230  CURVES  AND  SURFACES  [Chap.  XIII. 

through    0   meets   the  curve    in    A',  points,  consequently  counts 
for  -~{k^~l)   bisecants   through    0.     Similarly,   the   generator 

A  =  constant,  which  passes  through  (0,  0,  1,  0),  meets  the  curve 

k 
in  A'2  points  and  counts  for   -~  {k^  —  1)  bisecants.     The  number  of 

api^arent  double  points  is  the  sum  of  these  two  numbers. 

184.    Space  cubic  curves.* 

Theorem  I.  Tliroiigh  any  six  given  points  in  space,  no  four  of 
which  lie  in  a  plane,  can  he  passed  one  and  only  one  cubic  curve. 

Let  Pj,  •••,  Pq  be  the  given  points.  The  five  lines  connecting  P^ 
to  each  of  the  remaining  points  uniquely  determine  a  quadric 
cone  having  Pi  as  vertex.  Similarly,  the  lines  joining  Pj  to  each 
of  the  other  points  define  a  quadric  cone  having  P,  as  vertex. 
These  two  cones  intersect  in  a  composite  curve  of  order  four,  one 
component  of  which  is  the  line  P1P2,  since  it  lies  on  both  cones. 
The  residual  is  a  curve  of  order  three.  This  curve  cannot  be  com- 
posite, for  if  it  were,  at  least  one  component  would  have  to  be  a 
straight  line  common  to  both  cones.  But  that  would  require  that 
the  cones  touch  each  other  along  P1P2,  which  would  count  for  two. 
The  residual  intersection  would  in  that  case  be  a  conic  passing 
through  P3,  •••,  Po-  But  this  is  impossible  as  it  was  assumed  that 
the  points  P3,  —,  P«  do  not  lie  in  a  plane.  No  other  cubic  curve 
can  be  passed  through  the  given  points,  for  every  such  curve  would 
have  seven  intersections  with  the  two  cones  (the  vertex  counting 
for  two).  Hence  it  would  lie  on  their  curve  of  intersection,  wliich 
is  impossible,  since  the  complete  intersection  is  of  order  four. 

Theorem  II.  A  space  cubic  curve  lies  on  all  the  qnadrics  of  a 
bundle. 

For,  let  Pj,  •••,  P-  be  seven  given  points  of  the  curve.  Every 
quadric  through  these  points  has  2-3  +  1  points  in  common  with 
the  curve  and  consequently  contains  the  curve  (Art.  177).  But 
through  the  given  points  pass  all  the  quadrics  of  a  bundle  (Art. 
136),  which  proves  the  theorem. 

Not  all  the  quadrics  of  this  bundle  can  be  singular,  for  if  so,  at 

*  Unless  otherwise  stated,  it  will  be  assumed  in  the  following  discussion  that 
the  curve  is  non-composite. 


Arts.  183,  184]  SPACE   CUBIC   CURVES  231 

least  one  of  them  would  be  composite  (Art.  131)  and  still  contain  the 
curve.    This  is  impossible,  since  the  given  curve  is  not  a  plane  curve. 

The  symbol  (Art.  183)  of  a  space  cubic  curve  on  a  non-singular 
quadric  is  [2,  IJ  or  [1,  2],  since  such  symbols  as  [0,  3]  and  [3,  0] 
simply  define  three  generators  belonging  to  the  same  regulus. 

The  forms  of  /(A,  /x)  corresponding  to  these  symbols  are 

(a,X^  +  2  a,\  +  a,),M  +  (b,X  +2b,\  +  b,)  =  0,  (20) 

(aV^  +  2  aV  +  a',) A  +  (?> V  +  '-'  '^>  +  '^'2)  =  0.  (20') 

Conversely,  every  irreducible  equation  of  this  form  will  define 
a  cubic  curve  on  the  quadric. 

Since  these  equations  have  six  homogeneous  coefficients,  five  in- 
dependent linear  conditions  are  sufficient  to  determine  a  curve  of 
either  system.  Hence  through  any  five  points  on  a  given  non- 
singular  quadric  can  be  drawn  two  cubics,  one  of  each  symbol. 
Some  of  these  cubics  may  be  composite. 

From  the  formula  of  Art.  183  it  follows  that  on  a  given  non- 
singular  quadric  two  cubics  having  the  same  symbol  intersect  in 
four  points,  while  two  cubics  having  different  symbols  intersect 
in  five  points. 

Theorem  III.     Every  space  cubic  curve  is  rational. 

Let  the  parametric  equations  of  a  non-singular  quadric  through 
the  given  cubic  be  reduced  to  the  form  (19).  The  equations  of 
the  curve  in  A,  (x  are  of  the  form  (20)  or  (20').  In  (20),  let  X=t, 
solve  for  /x  in  terms  of  t,  and  substitute  the  values  of  A  and  of  /x 
in  terms  of  t  in  (19). 

The  resulting  equations  reduce  to  the  form 

X,  =  a^^f  +  a.^f'  +  a  J  -f  a-,,         i  =  1,  2,  3,  4.  (21) 

These  are  the  parametric  equations  of  the  curve  (Art.  178).  Since 
the  curve  is  by  hypothesis  of  order  three,  to  each  value  of  t  cor- 
responds a  definite  point  on  the  curve,  and  conversely. 

Since  the  cubic  (21)  does  not  lie  in  a  plane,  the  determinant 
I  a-j.  I  ^  0.  The  parametric  equations,  referred  to  the  tetrahedron 
defined  by 

Xi  =  a^ox'i  +  aiix'2  +  a.ax'j  +  a^^x^,         i  =  1,  2,  3,  4, 
are,  after  dropping  the  primes, 

x^  =  ^,     0^2  =  f",     X3  =t,     X4  =  1.  (22) 


232  CURVES  AND  SURFACES  [Chap.  XIIL 

From  (22),  the  intersections  of  the  curve  with  the  plane  '2,u-x-  =  0 
are  defined  by  the  roots  of  the  equation 

?/l«3  -f  M2«2  4-  Ust  +  »4  =  0.  (23) 

The  condition  that  the  pla.ne  is  an  osculating  plane  is  that  the 
roots  of  (23)  are  all  equal  (Art.  180).  It  follows  that  the  coor- 
dinates of  the  osculating  plane  at  the  point  whose  parameter  is  t 
may  be  expressed  in  the  form 

Mj  =1,     7(2  =  —  3  t,     u^  —  3  t',     Ui  =  —  f. 

These  equations  are  called  the  parametric  equations  of  the  cubic 
curve  in  plane  coordinates. 

The  condition  that  the  osculating  plane  at  the  point  whose 
parameter  is  t  passes  through  a  given  point  (//)  in  space  is  that  t 
is  a  root  of  the  equation 

lUf  -3y,t'  +  3y,t-y,=0.  (24) 

Since  this  equation  is  a  cubic  in  t,  it  follows  that  the  cubic  curve 
is  of  class  three. 

We  shall  now  pfove  the  following  theorem: 

Theorem  IV.  TJie  points  of  contact  of  the  three  osculating  jyJanes 
to  a  cubic  curve  throv/jh  an  arbitrarij  point  P  lie  in  a  plane  passing 
through  P. 

Let  %a-Xi  =  0  be  the  plane  passing  through  the  points  of  oscu- 
lation of  the  three  y^lanes  passing  through  any  given  point 
P  =  (?/).  The  parameters  of  the  points  of  osculation  of  the  three 
osculating  planes  through  {y)  are  the  roots  of  (24).  The  roots  of 
(24)  must  also  satisfy  the  equation 

ttii?  -\-  a^'^  +  aji  +  a^  =  0, 
hence 


'■^  Ik      -  V\ 

From  these  conditions  it  follows  that  2a,.V,  =  0,  so  that  (?/)  lies 
in  the  plane  of  the  points  of  osculation. 

By  the  method  of  Art.  179  the  equation  of  the  developable  sur- 
face of  the  cubic  curve  is  found  to  be 

(X^X^       X^pt'^j         4(^X2         X'^X'i^jyX^         3^2X4^=  U. 


Art.  184]  SPACE  CUBIC  CURVES  233 

This  is  also  the  condition  that  equation  (24)  has  two  equal  roots. 
From  this  equation  it  follows  that  the  rank  of  the  cubic  curve  is 
four  (Art.  179). 

It  was  stated  without  proof  in  Art.  133  that  the  basis  curve  of 
a  pencil  of  quadrics  of  characteristic  [22]  is  a  cubic  and  a  bi- 
secant ;  it  was  also  stated  that  the  basis  curve  of  a  pencil  of  char- 
acteristic [4]  is  a  cubic  curve  and  a  tangent  to  it.  We  shall  now 
prove  these  statements. 

It  was  shown  in  Art.  132  that  the  [22]  pencil  of  quadrics  is 
defined  by  the  two  surfaces 

x^^  +  2  x^x^  =  0,     x^^  +  2  x^x-i  =  0. 

These  quadrics  intersect  in  the  line  x^^O,  Xo  =  ^  and  the  space 
cubic  whose  parametric  equations  can  be  found  by  putting  Xi  =  1, 
Xi  =  2t  in  the  equations  of  the  surfaces,  in  the  form 

It  intersects  the  line  Xi  =  0,  x.,  =  0  in  the  two  points  (0,  0,  0,  1), 
(0,0,1,0). 

Similarly,  it  was  seen  that  a  pencil  of  characteristic  [4]  is 
defined  by  the  surfaces 

The  basis  curve  of  this  pencil  consists  of  the  cubic 

and  of  the  line  0:2  =  0,  x^  =  0  which  touches  it  at  (1,  0,  0,  0). 
If  in  the  parametric  equation  (20)  of  a  cubic  we  replace  X  by 

— ,  and  /x  by  -^,  we  determine  as  the  projecting  cone  from 
x^  x^ 

(0,  0,  1,  0)  a  cubic  cone  with  a  double  generator.     It  follows  that 

the  projecting  cone  of  the  cubic  is  intersected  by  a  plane  in  a 

nodal  or  cuspidal  plane  cubic  curve.     We  shall  now  prove  the 

converse  theorem. 

Theorem  V.  Any  nodal  or  cuspidal  plane  cubic  curve  is  the 
projection  of  a  space  cubic. 

Let  the  plane  of  the  cubic  be  taken  as  x^  =  0,  and  the  node  or 
cusp  at  (0,  1,  0,  0).     The  equation  of  the  curve  is  of  the  form 

XtittffCi  +  2  a^x^x^  +  anX^)  +  b^x^Xi  +  2  b^x^x^  +  b.,x^^  =  0. 


234  CURVES   AND   SURFACES  [Chap.  XIII. 

By  dividing  this  equation  by  x^  and  replacing  x, :  x^  by  \,  x^ :  x^ 
by  IX,  we  obtain  equation  (20)  of  a  space  cubic  curve  of  which  the 
given  curve  is  the  projection. 

Theorem  VI.  Amj  plane  nodal  cubic  curve  has  three  points  of 
inflexion  lying  on  a  line. 

If  a  space  cubic  is  projected  from  any  point  {y)  upon  a  plane, 
the  osculating  planes  from  {y)  will  be  cut  by  the  plane  of  projec- 
tion in  the  inflexional  tangents  of  the  image  curve  and  the  points 
of  osculation  will  project  into  the  points  of  inflexion.  From  the 
theorem  that  the  points  of  osculation  lie  in  a  plane  through  (?/)  it 
follows  that  the  points  of  inflexion  of  the  plane  cubic  lie  on  a  line. 

EXERCISES 

1.  Show  that  any  space  cubic  curve  and  a  line  which  touches  it  or  inter- 
sects it  twice  form  tlie  basis  curve  of  a  pencil  of  quadrics. 

2.  Show  that  a  composite  cubic  curve  exists,  through  which  only  one 
quadric  surface  can  pass. 

3.  Prove  that  the  osculating  planes  to  a  cubic  curve  at  its  three  points  of 
intersection  with  a  given  plane  (w)  intersect  at  a  point  in  (?(). 

4.  Show  tliat  if  a  cubic  curve  has  an  actual  double  point  or  a  trisecant  it 
must  lie  in  a  plane. 

5.  Obtain  all  the  Cayley-Salmon  numbers  for  the  proper  space  cubics. 

6.  Where  rau.st  the  vertex  of  the  projecting  cone  be  taken,  in  order  that 
the  plane  projection  of  a  proper  space  cubic  shall  have  a  cusp  ? 

7.  Show  that  the  projection  of  a  space  cubic  upon  a  plane  from  a  point 
on  the  curve  is  a  conic. 

8.  Show  that  the  cubic  curve  through  the  six  basis  points  of  a  web  of 
quadrics  determined  by  six  basis  points  lies  entirely  on  the  Weddle  surface 
(Art.  146). 

9.  Show  that  a  cubic  through  any  six  of  eight  associated  points  (Art. 
136)  will  have  the  line  joining  the  other  two  for  bisecant  (or  tangent). 

185.  Metric  classification  of  space  cubic  curves.  The  space  cubic 
curves  are  metrically  classified  according  to  the  form  of  their 
intersection  with  the  plane  at  infinity.  If  the  three  intersections 
are  real  and  distinct,  the  curve  is  called  a  cubical  hyperbola.  It 
has  three  rectilinear  asymptotes  and  lies  on  three  cylinders  all  of 
which  are  hyperbolic.  If  the  points  at  infinity  are  all  real  and 
two   are   coincident,    the   curve   is   called   a   cubical   hyperbolic 


Arts.  185,  186]  SPACE  QUARTIC   CURVES  235 

parabola.  It  has  one  asymptote,  and  lies  on  one  parabolic  cylin- 
der and  on  one  hyperbolic  cylinder.  If  all  three  of  the  points  of 
intersection  are  coincident,  the  plane  at  infinity  is  an  osculating 
plane.  The  curve  is  called  a  cubical  parabola.  It  has  no  recti- 
linear asymptote  and  lies  on  a  parabolic  cylinder.  Finally,  two 
of  the  points  of  intersection  may  be  imaginary.  The  curve  is 
now  called  a  cubical  ellipse.  It  has  one  rectilinear  asymptote 
and  lies  on  one  elliptic  cylinder.  An  interesting  particular  case 
of  the  cubical  ellipse  is  the  curve  called  the  horopter  curve  on 
account  of  its  part  in  the  theory  of  physiological  optics.  If  one 
looks  with  both  eyes  at  a  point  P  in  space,  the  eyes  are  turned  so 
that  the  two  images  fall  on  corresponding  points  of  the  retinae. 
The  locus  of  the  points  in  space  whose  images  fall  on  correspond- 
ing points  is  a  horopter  curve  through  the  point  P. 

186.   Classification  of  space  quartic  curves.* 

Theorem  I.  Every  space  q^iartic  curve  lies  on  at  least  one  quad- 
He  surface. 

For,  through  any  nine  points  on  the  curve  a  quadric  surface 
can  be  passed.  This  surface  must  contain  the  curve,  since  it  has 
2  X  4  -f  1  points  in  common  with  it  (Art.  177). 

If  a  quartic  curve  lies  on  two  different  quadrics  A  =  0,  B=  0, 
it  is  called  a  quartic  of  the  first  kind.  A  quartic  of  the  first  kind 
is  the  basis  curve  of  a  pencil  A  —  XB  =  0  of  quadrics.  Not  all 
the  quadrics  of  this  pencil  are  singular,  since  in  every  singular 
pencil  are  some  composite  quadrics.  Composite  quadrics  are  im- 
possible in  this  case,  since  the  curve  does  not  lie  in  a  plane.  The 
symbol  of  the  curve  on  any  non-singular  quadric  on  which  it 
lies  is  [2,  2],  since  each  generator  of  one  quadric  will  intersect 
the  other  quadric  defining  the  curve  in  two  points. 

A  quartic  having  the  symbol  [1,  3]  cannot  lie  on  two  different 
quadrics,  nor  can  it  lie  on  a  quadric  cone,  since  every  generator 
would  have  to  cut  the  curve  in  the  same  number  of  points.  The 
[1,  3]  curve  is  called  a  quartic  of  the  second  kind. 

It  follows  from  Arts.  132  and  184  that  except  in  the  cases  of 
the  characteristics  [1111],  [112],  [13],  the  basis  curve  of  a  pencil 

*  See  footnote  of  Art.  184. 


236  CURVES  AND  SURFACES  [Chap.  XIII. 

of  quadrics  is  composite.  It  will  now  be  shown  that  in  these 
three  cases  the  basis  curve  is  not  composite,  that  in  the  case 
[1111]  the  basis  curve  has  no  double  point,  that  in  the  case  [112] 
it  has  a  node,  and  that  in  the  case  [13]  it  has  a  cusp  or  stationary 
point  (Art.  181).  That  the  basis  curve  is  not  composite  may  be 
seen  as  follows:  If  it  were,  one  component  would  have  to  be  a 
line  or  a  conic.  It  cannot  be  a  line,  for  the  line  would  have  to 
lie  on  every  quadric  of  the  pencil,  hence  pass  through  the  vertex 
of  every  cone  contained  in  the  pencil.  From  the  equations  of  the 
pencils  having  these  characteristics  (Art.  133)  it  is  seen  that  in 
each  case  there  is  at  least  one  cone  whose  vertex  does  not  lie  on 
the  basis  curve.  Moreover,  one  component  cannot  be  a  conic,  for 
the  quadric  of  the  pencil  determined  by  an  arbitrary  point  P  in 
the  plane  of  the  conic  would  contain  the  plane  of  the  conic,  and 
hence  be  composite ;  but  pencils  having  these  characteristics  have 
no  composite  quadrics.  It  will  now  be  shown  that  the  basis  curve 
of  the  pencil  [1111]  has  no  actual  node  or  cusp.  It  will  be  called 
the  non-singular  quartic  curve  of  the  first  kind.  Suppose  the 
basis  curve  had  a  node  at  0.  The  projecting  cone  to  the  curve 
from  0  is  of  order  two.  The  quadric  of  the  pencil  through  an 
arbitrary  point  P  on  the  projecting  cone  contains  the  line  OP, 
since  it  has  three  points  in  common  with  it.  This  quadric  and  the 
cone  must  coincide,  since  they  have  a  quartic  curve  and  a  straight 
line  in  common.  Hence  the  cone  would  belong  to  the  pencil,  but 
this  is  impossible,  since  no  cone  of  the  pencil  [1111]  has  its  ver- 
tex on  the  basis  curve. 

From    the  equation   of    the  pencil  of   characteristic   [112]    it 
follows  that  the  vertex  (0,  0,  0,  1)  of  the  cone 

(A,  -  X,)x,^  +  {K  -  K)^-?  +  a^s'  =  0 

of  the  pencil  lies  on  the  basis  curve.  This  point  is  an  actual 
double  point  on  the  curve,  since  every  plane  through  it  has  two 
points  of  intersection  with  the  curve  coincident  at  that  point. 
All  the  quadrics  of  the  pencil  touch  the  plane  x-^  =  0  at  (0,  0, 0, 1); 
every  plane  through  either  of  the  distinct  lines  (A,,  —  A3)a;i'^ + 
(A2  —  \:i)x.^  =  0,  in  which  x^  —  0  intersects  the  cone  has  three  in- 
tersections with  the  curve  coincident  at  (0,  0,  0,  1).  These  two 
lines  are  tangents  at  the  node. 


Art.  186]  SPACE  QUARTIC   CURVES  237 

Finally,  the  vertex  of  the  cone 

of  the  [13]  pencil  is  a  double  point  on  the  basis  curve.  The  tan- 
gent lines  Xi  =  0,  x^  =  0  coincide.     The  double  point  is  a  cusp. 

The  parametric  equation  of  a  quartic  of  symbol  [2,  2]  has  the 
form 

(ta^  +  2  OiX  +  «o)^2  +  2(&o\'  +  2  6iX+&2V+  CoA2+2  c,\+c,  =  0.   (25) 

The  quartic  defined  by  (25)  is  the  intersection  of  the  quadric 
X1X2  —  x^x^  =  0  (Art.  1 83)  and  the  quadric 

aoO^s^  +  2  aiXs-Tj  -f  aj-r^^  +  2  boX^x^  +  4  b^x^Xz  +  2  62^2^4  +  (^0^1^  +  2  010:1X4 
'  +c,x,'  =  0.  (25') 

If  the  quartic  of  intersection  has  a  double  point  or  cusp,  we 
may  take  the  double  point  as  (0,  0,  0,  1),  and  a  cone  with  vertex 
at  that  point  for  one  of  the  quadrics  passing  through  it.  The 
parametric  equation  (25)  now  has  the  form 

(2  a,X  +  a,)ix?  +  2(b,\'  +  2  b,\),M  +  CoX^  =  0.  (26) 

If  in  (26)  we  put  X  =  fxt,  solve  for  t,  and  put  the  values  of  ft  and 
\  =  yd  in  equations  (19),  we  obtain  a  set  of  parametric  equations 
of  the  singular  quartic  curve  of  the  first  kind,  of  the  form 

x,  =  a,o^  +  a,,e  +  a,/  +  a  J  +  a,^,      i  =  1,  2,  3,  4 ;         (27) 

hence  the  nodal  and  cuspidal  quartics  are  rational. 

A  quartic  of  the  second  kind  can  be  expressed  parametrically 
in  terms  of  the  parameter  which  appears  to  the  third  degree  in  its 
parametric  equation,  hence  the  quartics  of  the  second  kind  are  also 
rational.     Rational  curves  will  be  discussed  later  (Art.  188). 

Theorem  II.  Through  a  quartic  curve  of  the  second  kind  and 
any  two  of  its  trisecants  can  be  passed  a  non-composite  cubic  surface. 

For,  through  nineteen  points  in  space  a  cubic  surface  can  be 
passed  (Art.  161).  Choose  thirteen  on  the  quartic  curve,  one  on 
the  trisecant  g,  one  on  the  trisecant  g',  not  on  the  curve,  and  four 
others  in  space,  not  in  a  plane  nor  on  the  quadric  on  which  the 
quartic  lies.  The  quartic  curve  and  the  lines  g  and  g'  must  lie  on 
the  non-composite  cubic  surface  determined  by  these  nineteen 


238  CURVES   AND   SURFACES  [Chap.  XIII. 

points  as  well  as  on  the  quadric  containing  the  regulus  of  tri- 
secants,  hence  together  they  form  the  complete  intersection  of  the 
cubic  and  the  quadric. 

187.  Non-singular  quartic  curves  of  the  first  kind.  Two  quartic 
curves  of  the  first  kind  lying  on  the  same  quadric  intersect  in 
eight  points  (Art.  183) ;  these  points  ai-e  eight  associated  points 
defining  a  bundle  (x\rt.  136),  since  they  lie  on  three  distinct 
quadrics  not  having  a  curve  in  common. 

The  number  of  apparent  double  points  of  a  non-singular  quartic 
Ci  of  the  first  kind  is  two.  For  each  bisecant  of  C^  through  an 
arbitrary  point  P  is  a  generator  of  the  quadric  of  the  pencil  hav- 
ing C4  for  basis  curve  which  passes  through  P.  Conversely,  each 
generator  of  every  quadric  through  C4  is  a  bisecant. 

Of  the  Cayley-Salmon  numbers  we  now  have  m  =  4,  h  =  2, 
/8  =  0,  H=  0.  It  also  follows  from  the  definition  that  G  =  v 
=  to  =  0,  hence  from  the  formulas  of  Art.  182  we  have 

m  =  4,  n  =  12,  r  =  8,  H=0,  h  =  2,  G  =  0,  g  =  38,  a  =  16,  (3  =  0, 
v  =  0,    0)  =  0,    x  =  16,    y  =  8,  J)  =  1. 

Theorem  I.  Through  any  bisecant  of  a  non-singular  space 
quartic  curve  of  the  first  kind  can  be  drawn  four  tangent  planes 
to  the  curve,  besides  those  having  their  poiyit  of  contact  on  the  given 
bisecayit. 

Let  the  given  bisecant  be  taken  as  .t,  =  0,  x^  =  0  and  the  quadric 
of  the  pencil  containing  it  as  x^Xi  —  x^x^  =  0.  Let  another  quadric 
of  the  pencil  be  determined  by  (25').  Any  plane  of  the  pencil 
Xi  =  mx^  intersects  C4  in  two  points  on  .x,  =  0,  x^  =  0  and  in  two 
other  points  determined  by  the  roots  of  the  quadratic  equation  in 
X2  '■  x^ 

x^  {a^ni^  +  2  a{tn  +  a,)  +  2  XM^  {bfpn"^  -\-  2  b^m  +  62) 
-f  x^  (cqWi^  +  2  Ci?«,  -\-  C2)  =  0. 

The  planes  determined  by  values  of  m  which  make  the  roots  of 
this  equation  equal  are  tangent  planes.     The  condition  on  m  is 

4(6om2+2  6im  +  &2)^-4(aom2+2  a^m  +  a.^{cam''-\-2  c{)n-\-c^  =  Q.    (28) 

Since  this  equation  is  of  the  fourth  degree,  the  theorem  is 
established. 


Art.  187]  QUARTICS   OF   THE   FIRST   KIND  239 

Theorem  II.  An  arbitrary  tangent  to  a  non-singular  quartic  of 
the  first  kind  intersects  four  other  tangents  at  points  not  on  the  curve 

This  is  a  particular  case  of  Theorem  I,  since  a  tangent  is  a 
bisecant  whose  points  of  intersection  with  the  curve  coincide. 

Theorem  III.  The  cross  ratio  of  the  four  tangent  planes  through 
any  bisecant  is  the  same  number  for  every  bisecant  of  the  curve. 

Two  cases  are  to  be  considered,  according  as  the  two  given 
bisecants  intersect  on  C^  or  not.  Let  g,  g'  be  two  bisecants 
through  a  point  P  on  C^,  but  not  lying  on  the  same  quadric  of 
the  pencil.  Let  the  equation  of  the  quadric  of  the  pencil  through 
C4  which  contains  g  be  reduced  to  the  form  x^x^  —  x^Xi  =  Q  in  such 
a  way  that  the  equations  of  g  are  x^  =  0,  054  =  0  and  the  points  of 
intersection  of  g'  with  C4  are  (0,  0,  1,  0)  (0,  0,  0,  1).  In  (25')  we 
now  have  a^  =  0,  c,  =  0,  and  also  in  (28).  The  points  of  inter- 
section not  on  g'  of  a  plane  x^  =  nx^  and  C4  are  determined  by  the 
roots  of  the  equation 

2  {c^n-  +  b.m)  x.^  +  (c^^n-  +  4  61/*  -f-  a,)  x^x^  +  2  {b^n  +  a^)  x^^  =  0. 

The  parameters  rii,  Wji  '>hj  ^h  0^  the  four  tangent  planes  are  roots 
of  the  equation 

(CqW^  +  4  b^n  +  aof  —  16  {bf^n  +  a^)  {c{n^  +  &2'0  =  0. 

The  cross  ratio  of  the  four  roots  of  this  equation  is  equal  to  the 
cross  ratio  of  the  roots  of  (28)  (when  Uq  =  03  =  0),  since  the  two 
equations  can  be  shown  to  have  the  same  invariants.* 

To  prove  the  theorem  when  g,  g'  intersect  at  P  on  C4  and 
lie  on  the  same  quadric  through  C4,  consider  any  third  bisecant 
g"  of  C4  through  P.  The  cross  ratios  on  g  and  on  g'  are  each 
equal  to  that  on  g". 

This  completes  the  proof  of  the  first  case. 

To  prove  the  theorem  when  the  two  bisecants  do  not  intersect 
on  Ci,  consider  a  third  bisecant  connecting  a  point  of  intersection 
on  the  first  with  a  point  of  intersection  on  the  second.  The 
cross  ratio  on  each  of  the  given  lines  is  equal  to  that  on  the 
transversal. 

This  cross  ratio  is  called  the  modulus  of  the  quartic  curve. 

*  Burnside  and  Panton :  Theory  of  Equations,  3d  edition,  p.  148,  Ex.  16.  It 
will  be  found  that  I  and  J  have  the  same  values  for  each  equation. 


240  CURVES  AND   SURFACES  [Chap.  XIII. 

The  projecting  cone  of  C\  from  a  point  on  it  is  a  cubic  cone. 
The  section  of  this  cone  made  by  a  plane  not  passing  through  the 
vertex  is  a  cubic  curve.  Conversely,  any  plane  cubic  curve  is  the 
projection  of  a  space  quartic  curve  of  the  first  kind.  Consider 
the  cubic  curve  in  the  plane  x^  =  0.  It  is  no  restriction  to  choose 
the  triangle  of  reference  with  the  two  vertices  (1,  0,  0,  0),  (0,  1, 
0,  0)  on  the  curve.  The  most  general  cubic  equation  in  x^,  x^,  Xi, 
but  lacking  the  terms  x^,  x.^,  may  be  written  in  the  form 

2  a^x.^Xi  +  a^x^x^  +  2  h^^Xi  +  4  b^x^x^x^  +  2  h^x^x^  -\-  c^x^Xi 
+  2  c^x^x^  +  c^x^  =  0. 

But  this  is  exactly  the  result  of  projecting  (Art.  175)  from  the 
point  (0,  0,  1,  0)  the  curve  (25)  for  the  case  ciq  =  0,  that  is,  when 
the  quartic  curve  passes  through  (0,  0,  1,  0). 

From  Theorem  III  it  now  follows  that  the  cross  ratio  of  the 
four  tangents  to  any  non-singular  cubic  curve  from  a  point  on  it, 
not  counting  the  tangent  at  the  point,  is  constant. 

It  was  seen  that  every  non-singular  quartic  lies  on  four  quadric 
cones  whose  vertices  (Art.  133)  are  the  vertices  of  the  tetrahedron 
self-polar  as  to  the  pencil  of  quadric  surfaces  on  which  the  curve 
lies  (Art.  112).  Let  t,  t'  be  two  distinct  tangents  of  C^  which 
intersect  in  a  point  P.  The  plane  tt  determined  by  t,  t'  touches  C^ 
in  the  points  of  contact  T,  T  of  t,  t',  respectively.  The  following 
properties  will  now  be  proved  : 

(1)  The  line  I  =  TT'  is  a  generator  of  a  quadric  cone  on  which 
Ci  lies. 

(2)  The  plane  tt  is  a  tangent  plane  to  this  cone  along  I. 

(3)  The  point  P  lies  in  the  face  of  the  self-polar  tetrahedron 
opposite  to  the  vertex  through  which  I  passes. 

The  plane  n  cuts  the  pencil  of  quadric  surfaces  on  which  C4  lies 
in  a  pencil  of  conies  touching  each  other  at  T  and  T.  One  conic 
of  this  pencil  consists  of  the  line  I  counted  twice,  hence  Z  is  a 
generator  of  a  cone  of  the  pencil  and  tt  is  its  tangent  plane.  More- 
over, I  is  the  polar  line  of  P  as  to  the  pencil  of  conies,  hence  the 
vertex  of  the  cone  and  the  point  P  are  conjugate  points.  Thus 
P  lies  in  that  face  of  the  self-polar  tetrahedron  which  is  opposite 
the  vertex  of  the  cone. 

If  TT  approaches  a  stationary  plane  (Art  181),  then   T,   T,  P 


Art.  187]  QUARTICS   OF   THE   FIRST    KIND  241 

approach  coincidence,  and  the  tangents  t,  t'  both  approach  I.  This 
occurs  at  every  point  in  which  C4  intersects  the  faces  of  the  self- 
polar  tetrahedron.     We  have  thus  the  following  theorem  : 

Theorem  V.  Tlie  points  of  contact  of  the  sixteen  stationary 
planes  (a  =  16)  of  a  non-singular  qaartic  curve  of  the  first  kind  lie 
in  the  faces  of  the  common  self-polar  tetrahedron.  The  planes  he- 
longing  to  the  points  in  each  face  pass  through  the  opposite  vertex. 

Referred  to  the  self-polar  tetrahedron,  the  equations  of  the 
quartic  are  (Art.  133) 

The  equation  of  the  developable  was  derived  in  Art.  179. 

The  section  of  the  developable  surface  by  the  plane  Xi  =  0  is 
the  quartic  curve  (o,^  =  a ■  —  a^), 

Ct24Cli3  Xi  X^     -\-  CI34I12  "^1  ■^2       1     Ct23(,Cll2^34     l"  ^IZ^^Uj'^i  X3     =  U 

counted  twice.  It  is  a  double  curve  on  the  developable.  It  is 
the  locus  in  the  plane  0:4  =  0  of  the  points  of  intersection  of  tan- 
gents to  C4.  A  similar  locus  lies  in  each  of  the  other  faces  of  the 
self-polar  tetrahedron.  Since  the  Cayley-Salmon  number  x  is  16, 
the  entire  locus  of  intersecting  tangents  to  C^  is  these  four  curves. 
Since  the  points  of  intersection  of  G^  with  the  faces  of  the 
self-polar  tetrahedron  are  the  points  of  contact  of  the  sixteen 
stationary  planes,  the  coordinates  of  these  points  are 

(±Va^,     ±Va3i,     ±Vai2,    0),     (±Va^,     ±  Va4i,    0,    ±Va|^), 

(±Va34,    0,     ±Va^,     ±Vai3),    (0,     ±  Va34,     ±Va42,  ±Va^). 

EXERCISES 

1.  Find  the  locus  of  a  point  P  such  that  the  two  bisecants  to  d  from  P 
coincide. 

2.  How  many  generators  of  each  quadric  through  d  are  tangent  to  the 
curve  ? 

3.  By  the  method  of  Art.  180  find  the  equations  of  the  stationary  planes. 

4.  Show  that  any  plane  containing  three  points  of  contact  of  stationary 
planes  will  pass  through  a  fourth.  How  many  distinct  planes  of  this  kind 
are  there  ? 

5.  Find  the  locus  of  a  point  P  such  that  the  plane  projection  of  d  from 
P  will  be  a  quartic  curve  with  one  double  point  and  one  cusp  ;  two  cusps. 


242  CURVES   AND   SURFACES  [Chap.  XIII. 

188.  Rational  quartics.  The  parametric  equations  of  any 
rational  quartic  may  be  written  in  the  form 

x^  —  a^^}  +  4  a-i^  +  6  a^J?  +  4  0,3^  +  a -4,     i  =  1,  2,  3,  4. 

The  parameters  of  the  points  of  intersection  of  the  curve  with 
any  plane  "^u-x^  =  0  are  the  roots  of  the  equation 

i^2«,a,o  +  4  t^'^u-aa  +  6  i^Su.a.-o  +  4  f^UiOis  +  '^u-a^  =  0. 

Let  ti,  <2,  is,  U  be  the  roots  of  this  equation.  From  the  formulas 
expressing  the  coefficients  in  terms  of  the  roots  we  have  at  once 

(<i  +  fo  +  ^3  +  ^4)  2a,o",  +  4  Saa"t  =  0, 
(tA  +  t,t,  +  t,t^  +  W3  +  U,  +  ^3^4)  2a,c?*t  -  6  %a,i>,,  =  0, 

(<i«2«3  +  t,UU  +  tUi  +  khQ  %a,,u^  +  4  ^a,,u^  =  0,         ^^^^ 
titnt-/-i%aiQi(^  —  2«,-4?^i  =  0. 

If  we  eliminate  n^  :  v^  :  "3  :  ^^4  from  these  four  equations,  we  obtain 
as  the  condition  that  t^,  •  •  •,  tt  are  the  parameters  of  four  coplanar 
points,  the  equation 

12  A,y,t/,  4-  3  J3(^^.,^3  +  tikfA  +  W,  +  ^,^3^ 

+  2  A,{t,t,  +  t,t,  +  ^1^4  +  U,  +  t,t,  +  ^3/4)         (30) 
+  3  A,{t,  +  t,  +  t,  +  t,)  +  12  Ao  =  0, 

in      which       ^0  =  |  f'llf*22«33f'44  1?      -^ll    =   1  «10«22'^33f'44  |j      ©tC.  If      <i  =  <2 

=  ^3  =  ^4  in  (29),  the  corresponding  point  will  be  a  point  of  con- 
tact of  a  stationary  plane.  Hence  there  are  four  points  of  con- 
tact of  stationary  planes.  These  four  points  are  defined  by  the 
equation 

A,t'  +  A,fi  +  A,f  +  A,t  +  A  =  0.  (31) 

Theorem.  If  a  quartic  airve  has  a  double  jwint,  the  parameters 
of  the  j)oints  of  contact  of  the  stationary  2)lan.es  are  harmonic. 

Let  P  be  the  double  point  and  let  ty,  t,  be  the  values  of  the  para- 
meter at  P.  Since  P  is  coplanar  with  any  other  two  points  on 
the  curve,  equation  (30)  is  satisfied  independently  of  the  values 
of  ^3  and  ^4.     Thus  ^1,  to  must  satisfy  the  conditions 

12  A,tfz  +  3  .-43  (t,  +  Q  +  2  ^2  =  0, 

3  AU2  +  2  A,  (/,  -I-  t,)  +  3  A,  =  0,  (32) 

2  A^t^  -f  3  ^1  («i  -f  t^)  -f  12  ^0  =  0. 


Art.  188]  RATIONAL  QUARTICS  243 

These  equations  are  compatible  only  when  the  determinant 
vanishes,  thus  ^.^  ^      ^^        ^^ 

3  A3    2  ^2      3  ^1  =  0. 
2.4,    3  A,    12  Ao 

But  this  is  the  condition  that  the  roots  of  (31)  are  harmonic* 

The  condition  that  the  double  point  is  a  cusp  is  t^  =  U.     In  this 
case  equations  (32)  are  replaced  by  the  quadratic  equations 
6  Af-  +  3  A^t  +  A^  =  0,     3  A^""  +  4  ^2^  +  3  ^i  =  0, 
A«2  +  3  ^i«  +  6  4,  =  0. 

But  these  are  the  conditions  that  (31)  has  a  triple  root.  Hence,  on 
a  cuspidal  quartic,  tliree  of  the  points  of  contact  of  stationary 
planes  coincide  at  the  cusp.  There  is  in  this  case  only  one  proper 
stationary  plane. 

Three  points  on  C^  are  collinear  if  their  parameters  t^,  to,  t^ 
satisfy  (30)  for  all  values  of  ^4.  The  necessary  conditions  are 
12  A,W,  +  3  A,(t,t,  +  t,t,  +  y,)  +  2  A,(t,  +  t,  +  ^3)  +  3  A,  =  0,  .33. 
3  A,t,y,  +  2  A,{t,t,  +  «2«3  +  y,)  +  3  A,(t,  +  U  +  t,)  +12  .lo  =  0.  ^  ^ 
If  the  curve  has  a  double  point,  the  parameters  t^,  to  of  the  double 
point  satisfy  these  conditions  for  every  value  of  ^3.  If  it  does 
not  have  a  double  point,  the  equations  (33)  are  satistied,  for  any 
given  value  of  ^3,  by  the  parameters  of  the  other  points  on  the 
trisecant  through  t. 

If  the  equations  resulting  from  (33)  by  putting  ?,  =  ^2  =  ^3  have 
a  common  solution  t',  the  curve  has  a  linear  inflexion  at  the  point 
whose  parameter  is  t'.  The  condition  that  tliese  equations  in  t' 
have  a  common  solution  is  exactly  the  condition  that  (31)  has  a 
double  root.  In  particular,  if  (31)  is  a  square,  the  curve  has  two 
distinct  linear  inflexions. 

EXERCISES 

1.   Obtain  the  Cayley-Salmon  numbers  for : 
(a)  the  nodal  quartic. 
(6)  the  cuspidal  quartic. 

(c)  the  general  quartic  of  the  second  kind. 

(d)  the  quartic  having  a  linear  inflexion. 

(e)  the  quartic  having  two  linear  inflexions, 

*  When  the  roots  of  a  quartic  equation  are  harmonic,  the  invariant  J  vanishes. 
See  Burnside  and  Pautou:  Theory  of  Equations,  4th  edition,  Vol.  1,  p.  150. 


244  CURVES  AND  SURFACES  [Chap.  XIII. 

2.  Show  that  every  [1,  !•]  curve  on  a  quadric  is  rational  and  can  have  no 
actual  double  point. 

3.  Show  that  every  rational  quartic  is  nodal,  cuspidal,  or  a  quartic  of  the 
second  kind. 

4.  Show  that  if  a  rational  quartic  does  not  have  a  cvisp  or  a  linear  in- 
flexion, its  parametric  equations  can  be  written  in  the  form 

.n  =  ((  +  1)4,     X2  =  (t  +  ay,     X3  =  t^,     X4  =  1. 
Find  the  values  of  a  for  which  the  curve  is  nodal. 

5.  Prove  that  if  a  quartic  has  a  single  linear  inflexion,  its  equations  can 
be  written  in  the  form 

xi  =  t\     X2  =  ^^     X3  =  {t+  l)^     Xi  =  1, 
and  if  it  has  two  distinct  linear  inflexions,  in  the  form 
Xi  =  t*,      X2  =  t^,      Xs  =  t,       0-4=  1. 

6.  Show  that  the  equations  of  a  cuspidal  quartic  can  be  written  in  the  form 

Xi  =  t*,      Xi  =  t^,      X3  =  t'^,      X4  =  1. 

7.  Show  that  the  tangents  at  the  points  of  contact  of  the  stationary  planes 
of  a  rational  quartic  are  in  hyperbolic  position  (Art.  120). 

8.  Show  that  through  any  point  P  on  a  rational  ([uartic  curve  pass  three 
osculating  planes  to  the  curve  besides  the  one  at  P,  and  that  the  plane  of  the 
points  of  contact  passes  through  P. 

9.  Determine  the  number  of  generators  of  a  quadric  surface  which  are 
tangent  to  a  [1,  3]  curve  lying  on  it. 

10.  Determine  the  number  of  generators  of  a  quadric  surface  which  are 
tangent  to  a  nodal  quartic  curve  lying  on  it. 

11.  Find  the  parametric  equations  in  plane  coordinates  of  the  curves  of 
Ex.5. 


CHAPTER   XIV 

DIFFERENTIAL  GEOMETRY 

In  this  chapter  we  shall  consider  some  of  the  properties  of 
curves  and  surfaces  which  depend  on  the  form  of  the  locus  in  the 
immediate  neighborhood  of  a  point  on  it.  Since  the  properties 
to  be  determined  involve  distances  and  angles,  we  shall  use  rec- 
tangular coordinates. 

I.   Analytic  Curves 

189.  Length  of  arc  of  a  space  curve.  The  locus  of  a  point  whose 
coordinates  are  functions,  not  all  constant,  of  a  parameter  u 

^=fi{u),     y:=f,(u),     z=f,(u)  (1) 

is  a  space  curve.  The  length  of  arc  of  such  a  curve  is  defined  as 
the  limit  (when  it  exists)  of  the  perimeter  of  an  inscribed  poly- 
gon as  the  lengths  of  the  sides  uniformly  approach  zero.  Curves 
for  which  no  such  limit  exists  will  be  excluded  from  our  discus- 
sion. 

By  reasoning  similar  to  that  in  plane  geometry  it  is  seen  that 
the  length  of  arc  s  from  the  point  whose  parameter  is  Wi  to  the 
point  whose  parameter  is  ^^  is 


This  equation  defines  s  as  a  function  of  u.  If  the  function  so 
defined  is  not  a  constant,  equation  (2)  also  defines  w  as  a  function 
of  s.     In  this  case  we  may  write  (1)  in  the  form 

x=x{s),     y  =  y(s),     z  =  z(s),  (3) 

in  which  s  is  the  parameter. 

Unless  the  contrary  is  stated,  we  shall  suppose  that  s  is  the 
parameter  in  each  case,  and  that  x,  y,  z  are  analytic  functions  of 

245 


246  DIFFERENTIAL   GEOMETRY  [Chap.  XIV. 

s  in  the  interval  under  consideration.  In  the  neighborhood  of 
{x{s),  y{s),  z(s)),  to  which  we  shall  refer  as  the  point  s,  we  have 

x" 
y^  =  x-\-  .a-'A.s  +  '—  (A.s)^  +  .. .  , 

y,  =  y-\-  y'^s  +  •(^'  (A.s')^-f  ...  ,  (4) 

in  which  a;'  =  — ,     a;"  =       ,  etc. 

rf.s  ds^ 

It  follows  from  ec^uation  (2)  that 

.r'2  +  y'^  +  z''  =  1.  (5; 

By  differentiating  equation  (5)  we  obtain 

x'x"  +  y'y"  +  z'z"  =  0.  (6) 

We  have  thus  far  supposed  that  the  second  member  of  (2)  was 
not  a  constant.  If  the  second  member  of  (2)  is  a  constant,  we 
have 

Curves  for  which  this  condition  (7)  is  satisfied  are  called  minimal 
f!urves.  They  will  be  discussed  presently.  It  will  be  supposed, 
except  when  the  contrary  is  stated,  that  the  curve  under  consider- 
ation is  not  a  minimal  curve. 

190.  The  moving  trihedral.  The  tangent  line  to  the  curve  at 
the  point  P  =  (x,  y,  z)  on  it  may  be  defined  as  the  limiting  posi- 
tion of  a  secant  as  two  intersections  of  the  line  with  the  curve 
approach  P. 

From  (4)  the  equations  of  the  tangent  at  P  are 

X—  X __  Y—  y _Z—  z  /ON 

x'     ~^~y^~     z'  ^^ 

Let  A,  fi,  V  be  the  direction  cosines  of  the  tangent,  the  direction 
in  which  s  increases  being  positive.     From  (8)  and  (5)  we  have 

A  =  a.',     /x=2/',     .  =  z\  (9) 


Art.  190]  THE   MOVING   TRIHEDRAL  247 

Tlie  plane  through  P  =  {x,  y,  z)  perpendicular  to  the  tangent  line 
is  called  the  normal  plane.     Its  equation  is 

x'{X  -x)^y\Y-y)-\-  z'  {Z-z)=  0.  (10) 

The  osculating  plane  at  P  is  the  limiting  position  of  a  plane 
through  the  tangent  line  at  P  and  a  point  P'  on  the  curve,  as  P' 
approaches  P.  We  chall  now  determine  the  equation  of  the 
osculating  plane. 

The  equation  of  any  plane  through  P  is 

A{X-  X)  +  B{Y -y)  ^  C{Z -z)  =  0. 

It  contains  the  tangent  (8)  if 

Ax'  +  By'  +  Cz'  =  0, 

and  will  be  satisfied  for  powers  of  As  up  to  the  third  (Eqs.  (4))  if 

Ax"  +  By"  +Cz"  =  ^. 

By  eliminating  A,  B,  C,  we  obtain,  as  equation  of  the  osculating 
plane  at  P, 

X-x       Y-y       Z-z 

x'  y'  z' 

x"  y"  z" 


=  0.  (11) 


The  line  of  intersection  of  the  osculating  plane  and  the  normal 
plane  is  called  the  principal  normal.  From  (10)  and  (11)  its 
equations  are  found  to  be 

X  —  x_  Y—y^Z—z  ,^2\ 

x"  y"  z"    '  ^     ' 

If  Xi,  /til,  vi  are  the  direction  cosines  of  the  principal  normal,  and 
if  we  put 

l  =  Va/'2+y'2^z"2,  (13) 

we  have 

K  =  px!',   ^,  =  py",    y,=pz".  (14) 

The  plane  through  P  perpendicular  to  the  principal  normal  is 
called  the  rectifpng  plane.     From  (12)  its  equation  is 

x'\X-x)  +  y"{Y-y)+z"{Z-z)  =  0.  (15) 


=  1.  (18) 


248  DIFFERENTIAL   GEOMETRY  [Chap.  XIV. 

The  intersection  of  the  rectifying  plane  and  the  normal  plane  is 

called  the  binormal.     From  equations  (10)  and  (15)  its  equations 

are 

X-x     ^      Y-y     ^      Z-z      ,  , 

y'z"-D"z'      z'x"-z"x'      x'y"-x"y'  ^     ' 

If  Xjj  ih.1  V2  are  the  direction  cosines  of  the  binormal,  we  have  the 
relations 

\,  =  p{,fz"  -y"z'\    ,x,  =  p{z'x"-z"x'),    v,  =  p{x'y"-x"y').     (17) 

The  trirectangular  trihedral  whose  edges  extend  in  the  positive 
directions  from  P  along  the  tangent,  principal  normal,  and  bi- 
normal is  called  the  moving  trihedral  to  the  curve  at  P  ={x,  y,  z). 

From  (9),  (14),  and  (17),  we  have 

\         ft         V 

It  follows  at  once  (Arts.  37,  38)  that  the  positive  directions  of 
the  coordinate  axes  can  be  brought  into  coincidence  .with  the 
positive  directions  of  the  moving  trihedral  at  the  point  P  by 
motion  alone,  without  reflexion.     Moreover,  we  have  (Art.  37) 

X     =  fJLlVi    —   V1JU2  ft     =  V1X2  —  A1V2,  V     =  A,/X2  —  Ao/Ai, 

Xi  =  fl^V    —  V2IJ',  fli  =  VoX    —  X2V,  Vi  =  X2JU.     —  X/U.2,  (19) 

X2  =   Vift     —  fX.iV  fX,  :=  XiV     —  VjX,  V-,  =   XfJii     —  Xifl. 

'  191.  Curvature.  The  curvature  of  a  space  curve  is  defined, 
like  that  of  a  plane  curve,  as  the  limit,  if  it  exists,  of  the  ratio 
of  the  measure  of  the  angle  between  two  tangents  to  the  length 
of  arc  of  the  curve  between  their  points  of  contact,  as  the  points 
approach  coincidence. 

Let  0  be  the  angle  between  the  tangents  to  the  curve  at  P  and 
P'.  The  direction  cosines  of  the  tangent  at  P  are  x',  y',  z'  (9), 
those  at  P'  are 

x'  +  x-"A.s  +  ...,    y'  -f  2/"As  +  -,    z'  +  2" As  -}-  -. 

From  Art.  5,  we  have 

sin^  A^  =  \  {y'z"  -  y"z'y  +  {z'x"  -  x'z"y  +  {x'y"  -  x"yy\  (As)2+  -., 


Arts.  191, 192] 


TORSION 


249 


the  remaining  terms  all  containing  higher  powers  of  As.  From 
(5)  and  (6)  the  coefficient  of  (A.s)-  reduces  to  x"^  +  y"^  +  z"\ 
Since 

lim-^H^^  =  l, 

we  have,  on  account  of  (13), 


da      p 


(20) 


as  the  expression  for  the  curvature  at  P.     The  reciprocal  of  the 
curvature  is  called  the  radius  of  cui-vature. 

If  -  =  0  at  a  point  P  on  the  curve,  the  tangent  at  P  has  three 
P 
points  of  intersection  with  the  curve  coincident  at  P;  hence  P  is 

a  linear  inflexion. 


192.  Torsion.  The  torsion  of  a  space  curve  is  defined  as  the 
limit,  if  it  exists,  of  the  ratio  of  the  angle  between  two  osculating 
planes  to  the  length  of  arc  between  their  points  of  osculation,  as 
the  points  approach  coincidence.  The  reciprocal  of  the  torsion 
is  called  the  radius  of  torsion  and  is  denoted  by  a. 

In  order  to  find  the  value  of  o-,  let  At  be  the  angle  between  the 
osculating  planes  at  the  points  whose  parameters  are  s  and  s  -}-  As. 
By  a  process  similar  to  that  of  Art.  191  we  obtain 

sin2  Ar  =  ]  (/x,v',  -  v,ii.\y  +  {v,X.',  -  \,u',y  +  ( V2  - 1^2^ 2)'  I  ( As)2  +  ..., 

the  remaining  terms  all  containing  higher  powers  of  As.     By  dif- 
ferentiating (17)  we  have 


X\  =  P-X,  +  p(y'z"' 
P 


fji,  +  p{z'x"'-x'z"'), 


It  follows  that 


p 


(21) 


X' 

y 

z 

fX^V2  —  V2fl2  =  P^x' 

x" 

y" 

z 

x'" 

y'" 

z 

with  similar  expressions  for  voA'o  —  Ajv'j  and  for  X^fx  ^  —  pL^K'^. 


250 


DIFFERENTIAL   GEOMETRY  [Chap.  XIV. 


If  we  substitute  these  values  in  the  above  expression  for  sin  At, 
pass  to  the  limit,  take  the  square  root,  and  assign  opposite  signs  to 
the  two  members,  we  obtain  the  result 


ds       cr 


X' 

y 

z 

=  -p' 

x" 

y" 

z' 

rJ" 

y'" 

z 

(22) 


which  is  the  formula  required.  Expand  the  determinant  of  equa- 
tion (22)  in  terms  of  the  elements  of  the  second  row,  replace 
x",  y",  z"  by  their  values  from  (14),  and  the  cofactors  of  these 
numbers  by  their  values  from  (21),  and  put  A1A2  +  /"•i/^2  +»')V2  equal 
to  zero,  since  the  principal  normal  and  binomial  are  orthogonal. 
By  performing  these  operations  we  simplify  (22)  to  the  form 


(23) 


193.    The  Frenet-Serret  formulas.     The  nine  equations 


A'  = 


Ai 


J  —  "1 


A',  =  -f^  +  ^W'i=      ^^^^ 


(^?)-  =  -C^?> 


(24) 


A  9  —         , 


H-2  —         ) 


/  Vi 

V2  =  -) 

cr 


are  called  the  Frenet-Serret  formulas. 

The  first  three  follow  at  once  by  replacing  X,  /x,  v  and  Aj,  /u,i,  vi 
by  their  values  from  (9)  and  (14). 

To  derive  the  last  three,  differentiate  the  identities 

V  +  fJ^2    +  vi  =  1,    AAo  +  fllX.2  -(-  VV2  =  0 

with  respect  to  s  and  substitute  for  A',  jx,  v  their  values  from  (24) 
which  we  have  just  established.     The  results  are 

AjA'j  -f  tHfJ^'i  +  V2v'2  =  0,    AA'2  +  /X/x'j  +  vv'2  =  0. 

From  these  equations  we  obtain,  after  simplifying  by  means  of  (19), 
A'2  =  SAi,  /x'2  =  8/xi,  v'2  =  Svi, 


Arts.  193,  194]         THE   OSCULATING  SPHERE  251 

8  being  a  factor  of  proportionality.  To  determine  its  value,  sub- 
stitute these  values  of  A'2,  yx^,  v'l  in  (23).     Since  \^  •\-  \x.^  +  ^i^  =  1, 

we  find  8  =  -.    The  last  three  equations  of  (24)  are  thus  established. 

To  find  the  values  A'l,  differentiate  the  identity  \  =  /^jv  —  vjyu. 
(19)  and  substitute  for  /x',  /,  /a',,  v'2  their  values  from  (24).     By 

(19)  the  result  reduces  to  the  form  A'i  =  — f-  +  -?j.  The  values 
of  /u,'i,  v\  are  found  in  the  same  way. 

194.  The  osculating  sphere.  The  sphere  which  has  contact  of 
the  third  order  with  a  curve  at  a  point  P  is  called  the  osculating 
sphere  of  the  curve  at  P.  To  determine  the  center  and  radius  of 
the  osculating  sphere  at  P^{x,  y,  z),  denote  the  coordinates  of 
the  center  by  (0:2,  y^,  ^2)  ^"^^  the  radius  by  R. 

The  equation  of  the  sphere  is 

(X  -  0^2)^  +  (  r-  2/2)2+  (^  _  ^^y  ^  ^2, 

This  equation  must  be  satisfied  by  the  coordinates  defined  by  (4) 
to  terras  in  (As)'  inclusive.  From  these  conditions  we  obtain  the 
following  equations 

{x  -  x,y  +  (y  -  y,y  +  (z  -z,y  =  r^, 

{x  -  x^)x'  +  {y-  y^)y'  +  (z  -  z.y  =  0, 

{X  -  x,)x"  +  (y  -  y,)y"  +(z-  z,)z"  +  1  =  0,  ^^""^ 

(x  -  x.y"  +{y-  yy"  +  (z  -  z,)z"'  =  0. 

By  solving  the  last  three  equations  for  x  —  x^,  y  —  y^,^  —  2^2  ^.nd 
simplifying  by  means  of  (21),  (22\  and  (24)  we  find 

X^  =  X-\-  p\i  —  p'crXo,  2/2  =  .'/  +  pP-i  —  p'fJ'H-iJ  Zo  =  Z  +  pvi—  p'crv2-      (26) 

If  we  substitute  these  values  of  a-,,  ^2'  ^2  ^^  the  first  of  equations 
(25)  and  simplify,  we  obtain 

R'  =  p^  +  (t'p'\  (27) 

Theorem.     TTie  condition  that  n  space  curve  lies  on  a  sphere  is 

p  +  (t{(t'p'  +  a-p")  =  0. 

If  a  given  curve  lies  on  a  sphere,  the  sphere  is  the  osculating 
sphere  at  all  points  of  the  curve  so  that  x^,  y^,  z^  and  R  are  con- 


252  DIFFERENTIAL   GEOMETRY  [Chap.  XIV. 

stants.     Conversely,  if  these  quantities  are  constants,  the  curve 
lies  on  a  sphere. 

To  determine  the  condition  that  the  coordinates  of  the  center 
are  constant,  differentiate  equations  (26)  and  simplify  by  means 
of  (24).     Since  Aj,  fj^,  v^  are  not  all  zero,  the  condition  is 

p  +  o-(cr'/3'  +  up")  =  0. 

By  differentiating  (27)  we  see  that  R  is  also  constant  if  this  equa- 
tion is  satisfied.     This  proves  the  proposition. 

195.  Minimal  curves.  We  have  thus  far  excluded  from  discus- 
sion those  curves  (Art.  189) 

x=f,{u),     y=f^{u),     z=f,{u), 
for  which 

Such  curves  we  called  minimal  curves.     A  few  of  their  properties 
will  now  be  derived. 

From  (28)  we  may  write 

dx  ,    .  di/  _      .  dz 

du        dii  du 

J  .(dx      .dy\      dz 

and  t{ — -—  1—^]=—- 

\du        dtij     du 

in  terms  of  a  parameter  t.     From  these  equations  we  deduce 
dx  dy  dz 

du    _      du      _  du 


2  2 

If  we  denote  the  value  of    these    fractions  by    <^(w),    solve    for 

— ,  — ,  —  and  integrate,  assuming  that  <^(m)  is  integrable,  we 
da    da     du 

find  that  the  equations  of  a  minimal  curve  may  be  written  in  the  form 

x  =  -  C{1-  t'^)(fi{u)du,       y=  f  ("(1  -f  t~)cf>{u)du,        z  =  Ct(fi{u)du^ 
2J  2J  J  ^29) 

in  which  i  is  a  constant  or  a  function  of  n.     If  t  is  constant,  the 

locus  (29)  is  a  line.     For,  let  k  be  defined  by  fc  =  |    cf>(u)du. 


Art.  195]  MINIMAL   CURVES  253 

lu  terms  of  k  we  obtain 

X  =  -^—  k  +  a?!,     y  =  ^(1  +  t'^)k  +  y^,     z  =  tk  +  z^, 

wherein  x-^,  ?/,,  z^^  are  constants  of  integration.     The  locus  of  the 
point  {x,  y,  z)  is  the  minimal  line  through  the  point  {x^,  y^,  z^ 

x—Xi        y-yx        z-Zx 


\-P      i(l  +  O  t 

2  2 

The  equation  of  the  locus  of  the  minimal  lines  through  any  point 
(Xi,  ?/,,  Zi)  is  found  by  squaring  the  terms  of  these  equations  and 
adding  numerators  and  denominators,  respectively,  to  be  the  cone 

{X.-X,Y+  y-y,Y^{z-Z,Y=^, 

having  its  vertex  at  (xi,  ?/i,  z^  and  passing  through  the  absolute. 
This  is  identical  with  the  equation  of  the  point  sphere  (Art.  48). 
If  t  is  not  constant,  but  a  function  of  ?t,  we  may  take  t  as  the 
parameter.  Let  u  =  \\i{t.),  and  let  ^{iC)(ln  ^=  <^{^{tj)\\i^ {pjdt  be  re- 
placed by  F{t)dt.     Equations  (29)  have  the  form 

x  =  ^^{l-t')F{t)dt,     y='^j{l  +  t')F{t)dt,     z=JtF{t)dL     (30) 

Jjet  f{t)  be  defined  by-^  =  i^(^).  By  integrating  equations  (30) 
by  parts  we  have 

y=^s^^m-u'J^+im+y,,        (31) 

dt^  dt 

^i>  yi>  ^\  being  constants.  The  equations  of  any  non-rectilinear 
minimal  curve  may  be  expressed  in  this  form. 

EXERCISES 

1.    The  curve 

X  =  a  cos  0,     y  —  a  sin  cj),     z  =  a(j> 

is  called  a  circular  helix.  Find  the  parametric  equations  of  the  curve  in 
terms  of  the  length  of  arc. 


254 


DIFFERENTIAL  GEOMETRY 


[Chap.  XIV. 


2.  At  an  arbitrary  point  of  the  helix  of  Ex.  1  find  the  direction  cosines 
of  the  tangent,  principal  normal,  and  binormal.  Also  find  the  values  of  p 
and  (T. 

3.  Find  the  radius  of  the  osculating  sphere  at  an  arbitrary  point  of  the 
space  cubic  x  =  t ,  y  =  t^,  z  —  t^. 

4.  Show  that  the  equations  of  a  curve,  referred  to  the  moving  trihedral  of 
a  point  P  on  it,  may  be  written  in  the  form 


+ 


z=-  —  + 
o  p(r 


2p      \ds)6p^ 
s  being  the  length  of  arc  from  P. 

5.  Discuss  the  equations  (31)  of  a  minimum  curve  in  each  of  the  follow- 
ing cases: 

C'^)  /(O  ^  quadratic  function  of  t. 
(6)  f(t)  a  cubic  function  of  t. 

II.  Analytic  Surfaces 

196.  Parametric  equations  of  a  surface.  The  locus  of  a  point 
(x,  y,  z)  whose  coordinates  are  analytic  real  functions  of  two  in- 
dependent real  variables  u,  v 

x=f^{u,v),   y=f2{u,v),   z=fs{u,v),  (32) 

such  that  not  every  determinant  of  order  two  in  the  matrix 


M  ^  ^3 

du  du  du 

df\  df,  % 

do  dv  dv 


(33) 


is  identically  zero,  is  called  an  analytic  surface.  The  locus  de- 
fined by  those  values  of  ri,  v  for  which  the  matrix  (33)  is  of  rank 
less  than  two  is  called  the  Jacobian  of  the  surface.  Points  on 
the  Jacobian  will  be  excluded  in  the  following  discussion. 

The  reason  for  the  restriction  (33)  is  illustrated  by  the  follow- 
ing example. 

Example.     Consider  the  locus 

X  =  u  +  V,    y  =(u  +  vy,    z  =(u  +  vy. 
For  any  given  value  t,  any  pair  of  values  ?«,  v  which  satisfy  the  equation 
u  -{-  V  =  t  define  the  point  (t,  t'-,  «').     The  locus  of  the  equations  is  a  space 
cubic  curve.     In  this  example  the  matrix  (33)  is  of  rank  one. 


Arts.  196-198]         TANGENT   PLANE.    NORMAL  LINE        255 

The  necessary  and  sufficient  condition  that  u,  v  enter /i,/2,/3  in 
such  a  way  that  x,  y,  z  can  be  expressed  as  functions  of  one  vari- 
able is  that  the  matrix  is  of  rank  less  than  two. 

197.  Systems  of  curves  on  a  surface.  If  in  (32)  u  is  given  a 
constant  value,  the  resulting  equations  define  a  curve  on  the  sur- 
face. If  u  is  given  different  values,  the  corresponding  curve 
describes  a  system  of  curves  on  the  surface.  Similarly,  we  may 
determine  a  system  of  curves  v  =  const.  The  two  systems  of 
curves  n  =  const.,  v  =  const,  are  called  the  parametric  curves  for 
the  given  equations  of  the  surface ;  the  variables  u,  v  are  called 
the  curvilinear  coordinates  on  the  surface. 

Any  equation  of  the  form 

<t>{u,  v)  =  c  (34) 

determines,  for  a  given  value  of  c,  a  curve  on  the  surface.  The 
parametric  equations  of  the  curve  may  be  obtained  by  solving 
(34)  for  one  of  the  variables  and  substituting  its  value  in  terms  of 
the  other  in  (32).  If  we  now  give  to  c  different  values,  equation 
(34)  determines  a  system  of  curves  on  the  surface. 

If  <f>(ii,  v)  =  c,  \p{u,  V)  =  c'  are  two  distinct  systems  of  curves  on 
the  surface,  such  that 

dcf)  dip      dcf>  5i/'      p> 

du  dv      dv  du 
by  putting  <^(?*,  v)  =  u',  \p  (n,  v)  =  v'  and  solving  for  u,  v  we  may 
express  x,  y,  z  in  terms  of  u',  v'.     This  process  is  called  the  trans- 
formation of  curvilinear  coordinates. 

198.  Tangent  plane.  Normal  line.  The  tangent  plane  to  a  sur- 
face at  a  point  P  on  it  is  the  plane  determined  by  the  tangents 
at  P  to  the  curves  on  the  surface  through  P. 

The  equations  of  the  tangent  lines  to  the  curves  u  =  const, 
and  V  =  const,  at  P  =  (x,  y,  z)  =  {u,  v)  are  (Art.  190) 
X-x      Y-y      Z-z 


dx 

dv 

dz_ 
dv 

X-x 

Y-y 

Z-z 

dx 
du 

dji 
du 

dz 
du 

256 


DIFFERENTIAL  GEOMETRY 


[Chap.  XIV. 


The  plane  of  these  two  lines  is 

X-x     T-y    Z-z 


dx 

dy 

dz 

dti 

du 

du 

dx 

dy 

dz 

dv 

dv 

dv 

=  0. 


(35) 


Let  V  =  </>(«)  be  the  equation  of  any  other  curve  on  the  surface 
through  (u,  v).     The  equations  of  its  tangent  lines  are 

X-x  Y-y  Z-z 


X 

dx  I   dx  dcj) 
du      dv  du 


dy       dy  d4> 
du      dv du 


dz       dz  d(l> 
du      dv  du 


This  line  lies  in  the  plane  (35)  independently  of   the   form    of 
<l>(u),  hence  (35)  is  the  equation  of  the  tangent  plane. 

The  normal  is  the  line  perpendicular  to  the  tangent  plane  at 
the  point  of  tangency.     Its  equations  are 

Y-v  Z-z 


X  —  X 


dy  dz      dz  dy      dz  dx  _  dx  dz 
du  dv      du  dv      du  do      du  dv 


dx  dy 
du  dv 


dy  dx 
du  dv 


We  shall  denote  the  direction  cosines  of  the  normal  by  X,  fx.,  v. 

Their  values  are 

r  _  1  fdy  dz  _  dz  dy'^ 

D\dudv      dudVj 


wherein 

'3?/  dz 


D'  = 


du  dv 


dz  dy^ 
du  dv 


_  _  1  /dz  dx  dx  dz 
D\du  dv      dudv 

_  _  1  fdx  dy  _  dy  dx 
D  \du  dv      du  dv 

dz  dx      dx  dz 


-^^    M-^~- 


dudv      dudv 


4- 


dx  dy 
du  dv 


dy  dx\^ 
du  dv) 


(36) 


(37) 


If  D  —  0,  the  tangent  plane  (35)  is  isotropic  (Art.  152),  and  the 
formula  for  determining  the  direction  cosines  of  the  normal  fails. 
We  shall  limit  our  discussion  to  the  case  in  which  D^O. 

The  equation  of  the  tangent  plane  may  be  written  in  the  form 

X{X-x)  +  ]J.{Y-y)^v{Z-z)=0. 


A.RT.  199]  DIFFERENTIAL  OF  ARC  257 

199.  Differential  of  arc.  Let  <j){u,  v)=0  be  the  equation  of  a 
curve  on  the  surface  (32).  The  differential  of  the  length  of  arc  of 
this  curve  is  given  by  the  formula  (Art.  189) 

ds^  =  dx^  -\-  dy^  +  dz^, 
in  which 

dx  =  -^  du  -\ — '-  dv,  dy  = -^  dxi  4- -^  dv,  dz  =  —  du  -\ dv, 

u  dv  ou  dv  du  dv 

and  the  differentials  du,  dv  satisfy  the  equation 

du  dv 

If  we  substitute  these  values  for  dx,  dy,  dz  in  the  expression  for 
ds  we  obtain 

ds^  =  Edu''  +  2  Fdudv  +  Gdv'^,  (38) 

in  which 


-=(ST-(IJ-(SJ' 


du  dv      du  dv      du  dv 


^=y  \dv)  \dv) 


Since  the  expression  <^{u,  v)  does  not  enter  explicitly  in  the  equa- 
tion (39),  the  expression  for  ds  has  the  same  form  and  the  coefft- 
cients  E,  F,  G  have  the  same  values  for  all  the  curves  passing 
through  P,  but  the  value  of  dv  :  du  depends  upon  the  curve  chosen. 
The  coefficients  E,  F,  G  are  called  the  fundamental  quantities  of 
the  first  order.     From  (37)  and  (39)  it  follows  that 

D^  =  EG-  F\ 

Let  C  be  a  curve  on  the'  surface  through  {u,  v)  and  let  ds  be  the 
element  of  arc  on  C.  The  direction  cosines  A,  fx,  v  of  the  tangent 
to  C  are 

._dx_dxdudxdv^  _dy  _dy  du      dy  dv 

ds      du  ds      dv  ds  ds      du  ds      dv  ds' 

_  dz  _  dz  du      dz  dv 
ds     du  ds      dv  ds 


258  DIFFERENTIAL   GEOMETRY  [Chap.  XIV. 

If  we  replace  ds  by  its  value  from  (38),  divide  numerator  and 

denominator  of  each  equation  by  da,  and  replace  dv :  du  by  k, 

we  have 

dx  ,,dx 

du         do 
A  ^  • 


VE  +  2Fk-\-Gk^' 


^  +  k^ 

du         dv  .,^^ 

IX  =                      ,  (40) 

VE-{-2Fk  +  Gk^  ^     ^ 

—  -I-  A-  — 
du         dv 

"  ~  ^W+YW+Gk^' 

It  follows  from  these  equations  that  at  a  given  point  on  the  sur- 
face the  tangent  line  to  a  curve  passing  through  the  point  is 
uniquely  deteiunined  when  the  value  of  the  ratio  dv  :  du  =  k  is 

known,  since  -^,  — ,  etc.,  are  fixed  when  the  point  (u,  v)  is  given. 
du    dv 

200.    Minimal  curves.     Each  factor  of  the  expression 

Eda^  +  2  Fdudv  +  Gdv\ 

when  equated  to  zero,  determines  a  system  of  curves  on  the  sur- 
face. Let  <^  (?/,  v)  du  +  ^^  (u,  v)  dv  be  such  a  factor.  By  equating  to 
zero  and  integrating  we  obtain  an  equation  of  the  form  f(u,  v,)=  c, 
in  which  c  is  a  constant  of  integration,  which  determines  a  system 
of  curves  on  the  surface. 

The  two  systems  of  curves  determined  in  this  way  are  minimal 
curves  (Art.  195),  since  the  differential  of  arc  of  every  curve  of 
each  system  satisfies  the  condition 

(7i-2  =  Edu^  +  2  Fdudv  -f-  Gdv''  =  0. 

This  equation  determines,  at  (u,  v),  two  values  of  the  ratio 
dv  :  du  =  k  which  define  two  imaginary  tangents  to  minimal 
curves.     The  two  tangents  coincide  at  points  for  which  D=  0. 

In  the  succeeding  discussion  we  shall  assume  that  minimal 
curves  are  excluded. 


Arts.  200-202]        RADIUS  OF  NORMAL  CURVATURE       259 

201.  Angle  between  curves.  Diiferential  of  surface.  The  angle 
between  the  tangents  to  the  curves  u  =  const.,  v  =  const,  is  deter- 
mined from  Art.  198  by  the  formula  (Art.  5) 


F       f  T  ,     •  ^EG  -  F'         D 

cos  oj  =  — ,  from  which  sm  w  = 


^'EG  -JEG  ^EG 

The  curvilinear  quadrilateral  whose  vertices  are  determined  by 
{u,  v),  (n  -j-  Am,  v),  {u,  v  -\-  Aw),  {u  +  A»,  v  +  Ay)  is  approximately 
a  parallelogram  such  that  the  lengths  of  the  adjacent  sides  are, 
from  (38)",  -VEdu,  -y/Gdv,  and  the  included  angle  is  w. 
Hence  we  have  in  the  limit  for  the  differential  of  surface 

dS  =  sin  (D^EGdudv  =  Ddudv. 

Let  C,  C  be  tw^o  given  curves  on  the  surface  through  a  point  P. 
We  shall  denote  the  differentials  of  n,  v,  s  on  C  by  dii,  dv,  ds  and 
the  differentials  of  u,  v,  s  on  C  by  8u,  Sv,  8s.  The  direction  co- 
sines A,  /A,  V  of  the  tangent  to  C  are  determined  by  replacing  k  in 
(40)  by  dv-.du;  similarly  the  direction  cosines  A',  [x,  v  of  the 
tangent  to  C  are  determined  by  replacing  Iz  by  Sy  :  Z^i. 

If  Q  is  the  angle  between  the  tangents  to  C  and  C  at  (w,  -y), 

oos«=XX'  +  m/+v/  = ^-^^^jj ^ (41) 

From  (41)  we  have  at  once  the  following  theorem: 

Theorem.  The  condition  that  tioo  directions  determined  by  the 
ratios  do  :  da,  hv  :  Sic  are  orthogonal  is 

EduSu  +  F(du8v  +  dvBu)  +  Gdv8v  =  0.  (42) 

202.    Radius  of  normal  curvature.    Meusnier's  theorem.     Let  ip 

be  the  angle  which  the  principal  normal  to  C  makes  with  the 
normal  to  the  surface.  Let  Ai,  fii,  vi  denote  the  direction  cosines 
of  the  principal  normal  and  ds  the  differential  of  arc  along  C. 
We  have,  from  (14) 

cos^  =  A,A  +  /.„.  +  v,v  =  p^A-  +  ^^-t-v-j, 
p  being  the  radius  of  curvature  of  C  at  (m,  v). 


(43) 


260  DIFFERENTIAL  GEOMETRY  [Chap.  XIV. 

But 

d'^x  _  d'^x  /d«Y      _    d'^x  du  dv  ,  d'^x/dvV      dx  dhi      dx  d?v 
ds^      du'^\dsj       ~'  dudvdsds      dv^\dsj       du  ds^      dvds"^' 

with  similar  expressions  for  — -, Substitute  these  values 

ds^     ds^ 

for  the  second  derivatives  in  the  equation  for  cos  if/.     Since  the 

normal  to  the  surface  is   perpendicular  to  the  tangents   to   the 

curves  u  —  const.,  v  =  const.,  we  have  the  relations 

-dx         By         dz      ^      -  dx      _  df/         dz      ^ 
ou         au         du  dv         do         dv 

If  we  replace  ds  by  its  value  from  (38),  the  equation  for  cos  \p  may 
be  reduced  to 

cos  if/  _  Ldu-  +  2  Mdudv  +  Ndv^ 
p     ~  Edu""  +  2  Fdudo  +  Gdv'^' 
wherein 

r  _  T  ^"^    ,    -  ^^y    ,    -  ^'^ 
d^r  dir         du^ 

31^1^^  +  ].^^  +  .^,  (44) 

dudv  dude         dudv 

dv^  dir         dv^ 

The  quantities  L,  M,  N  are  called  the  fundamental  quantities  of  the 
second  order  for  the  given  surface. 

The  second  member  of  equation  (43)  depends  only  on  (m,  v)  and 
the  ratio  dv  :  da  =  A;.  Consider  the  plane  section  of  the  surface 
determined  by  the  normal  to  the  surface  and  the  tangent  to  C. 
Such  a  section  is  called  a  normal  section.  Let  the  radius  of 
curvature  of  this  normal  section  at  (u,  v)  be  denoted  by  B.  From 
(43)  we  have 

li  ~  Edu}  +  2  Fdudv  +  Qdv'''  ^    ^ 

Rcosil/  =  p.  (46) 


and  hence 


The  result  expressed  in  equation  (46)  may  be  stated  in  the  follow- 
ing form,  known  as  Meusnier's  theorem  : 


Arts.  202-204]  CONJUGATE   TANGENTS  261 

Theorem.  Tlie  center  of  curvature  of  any  point  of  a  curve  on  a 
surface  is  the  projection  on  its  osculating  plane  of  the  center  of  curva- 
ture of  the  normal  section  tangent  to  the  curve  at  the  p)oint. 

203.  Asymptotic  tangents.  Asymptotic  curves.  The  two  tan- 
gents to  the  given  surface  at  (w,  v)  defined  by  the  equation 

Ldu}  +  2  Mdudv  +  Ndd"  =  0  (47) 

are  called  the  asymptotic  tangents  at  P. 

From  (45)  we  have  at  once  the  following  theorem : 

Theorem  I.  If  the  curve  C  on  the  surface  is  tangent  to  an  asymp- 
totic tangent  at  (u,  v),  then  either  the  osculating  plane  to  C  coincides 
with  the  tangent  plaiie  to  the  surface  or  C  has  a  linear  inflexion  at 
(u,  v). 

The  two  systems  of  curves  defined  by  the  factors  of  (47)  are 
called  the  asymptotic  curves  of  the  surface.  They  have  the  prop- 
erty that  their  tangents  are  the  asymptotic  tangents  to  the  sur- 
face.    We  have  the  further  theorems : 

Theorem  II.  If  a  straight  line  lies  on  a  surface,  it  coincides 
with  an  asymptotic  tangent  at  each  of  its  points,  hence  the  line  is  an 
asymptotic  curve. 

Theorem  III.  The  osculating  plane  at  each  point  of  a  real 
asymptotic  curve,  not  a  straight  line,  coincides  with  the  tayigent  plane 
to  the  surface  at  that  point. 

204.  Conjugate  tangents.  The  equations  of  the  tangent  planes 
at  P=  {x,  y,  z)  and  at  P'  =(a;  +Ax,  ?/  +  Ay,  z  -(-  Az)  on  the  surface 
are  (Art.  198) 

^X-x)-\-]l(Y-y)-\-v(Z-z)=0, 
(A  +  A\){X-x  -  Ax)  -{-(Ji  +  A]x){T-y-  Ay) 

+  (v-\-Av)iZ-z-Az)=0.  (48) 

Let  P'  approach  P  along  a  curve  whose  tangent  at  P  is  deter- 
mined by  k  =  dv  :  du.  We  shall  now  determine  the  limiting  posi- 
tion of  the  line  of  intersection  of  the  planes.     If  we  subtract  the 


262  DIFFERENTIAL  GEOMETRY  [Chap.  XIV. 

first  of  equations  (48)  from  the  second,  member  by  member,  and 
pass  to  the  limit,  we  have 

d\(X—  x)  +  dfL{  Y—y)  +  dv  {Z  —  z)+  ~Xdx  +  jldy  +  vdz  =  0. 

But  Adx  +  jldy  -\-  vdz  =  0,  sinr^e  the  normal  to  the  surface  at  P  is 

perpendicular  to  every  tangent  at  P.    Hence  the  limiting  position 

of  the  line  of  intersection  passes  through  P,  since  it  lies  in  the 

tangent  plane  at   P  and    in   the   plane    dk{X  —  x)  +  d'fj.^Y  —  y) 

+  dv(Z— ^)=0  through  P.     Let  the  point  {X,  Y,  Z)  on  the  line 

Sx  doc 

of  intersection  be  denoted  by  X—x-\-^x  —  x-\ 8m  -1 Sv,  etc. 

du  dv 

(Art.  199).     We  have 

d\(^-^8u  +  ^-^8v)  +  d-J^8u  +  ^Bv\+d-J^-^8u  +  ^-^Bv)  =  0. 
\du  dv     J  \du  dv      J  \da  dv      J 

If  we  replace  A,  ju.,  v  by  their  values  from  (36)  and  simplify,  this 
equation  reduces  to 

LduBu  +  M(du8v  -f-  dv8u)  +  NdvSv  =  0,  (49) 

which  determines  Bv  :  Bu  linearly  in  terms  of  dv :  du. 

Since  equation  (49)  is  symmetric  in  dv :  du  and  Bv :  8m,  it  follows 
that  if  a  point  P"  approaches  P  in  the  direction  determined  by 
8v :  Su,  the  limiting  position  of  the  line  of  intersection  of  the  tan- 
gent planes  at  Pand  P"  is  determined  by  dv:  du. 

Two  tangents  determined  by  dv :  du,  Bv :  Bu  which  satisfy  (49) 
are  called  conjugate  tangents. 

Theorem.  The  necessary  and  sufficient  condition  that  a  tangent 
coincides  with  its  conjugate  is  that  it  is  an  asymjytotic  tangent. 

For,  if  in  (49)  we  put  Bv  :  Bu  =  dv  :  du,  we  obtain  (47).  Con- 
versely, if  dv :  du  satisfies  (47)  and  Bv :  Bu  is  conjugate  to  it,  then 
dv:  du=  Bv:  Bu. 

205.  Principal  radii  of  normal  curvature.  In  order  to  determine 
the  maximum  and  minimum  values  of  R  in  equation  (45)  at  a 
given  point  (u,  v)  put  dv:  du  =k  and  differentiate  li  as  a  function 
of  Jc.  The  derivative  vanishes  for  values  of  k  determined  by  the 
equation 

(FN-  GM)k^  -(GL-  EN)k  +  (EM-  FL)  =  0.  (50) 


Arts.  205, 206]  LINES  OF  CURVATURE  263 

If  this  equation  is  not  identically  satisfied,  the  two  roots  k^,  k^  are 
real  and  distinct,  since  the  part  under  the  radical  may  be  expressed 
as  the  sum  of  two  squares. 

(GL  -  ENf  -  4:{FN-  GM){EM-  FL) 

=  4  —lEM  -FLy  +  [EN  -GL-^  (EM  -  FL)  J. 

One  root  will  determine  the  tangent  dv :  dii  such  that  the  normal 
section  through  it  will  have  a  maximum  radius  of  curvature  R^  and 
the  other  will  determine  the  normal  section  having  the  minimum 
radius  of  curvature  i?2. 

The  tangents  at  (u,  v)  determined  by  the  roots  of  (50)  are  called 
the  tangents  of  principal  curvature,  and  the  corresponding  radii 
Ml,  i?2  ^'^e  called  the  principal  radii  of  curvature.  To  determine 
the  values  of  R^  and  ^2  we  have  from  (45)  and  (50) 

L  +  kM^  M+kN^  ]_^ 
E  +  kF      F  +  kG      R 

By  eliminating  k  between  these  equations,  we  obtain  the  quad- 
ratic equation 

{LN-  M^)R'  -  {EN-  2FM  +  GL)R  +  EG-F'  =  0,       (51) 

whose  roots  are  R^  and  R^. 

The  expression \ is  called  the  mean  curvature  of  the  sur- 

Ri     R2 

face  at  (u,  v) ;  the  expression  —  •  —  is  called  the  total  curvature 

Ri     R2 

of  the  surface  at  (u,  v).     From  (51)  we  have 

Ri     R,  EG-F^        '  ^  "^ 

1         LN-  M- 


R1R2     EG-F^ 

206.  Lines  of  curvature.   If  in  (50)  we  put  k  =^dv:  du,  we  obtain 

{E3f-  FL)du-  -  (GL  -  EN)dudv  +  (FN  -  GM)do-  =  0.        (53) 

The  two  factors  of  this  equation  determine  two  systems  of  curves 
called  lines  of  curvature  of  the  surface.     If  the  two  directions  at 


264  DIFFERENTIAL  GEOMETRY         [Chap.  XIV. 

(u,  v)  of  the  lines  of  curvature  are  denoted  by  dv  :  du  and  8v  :  8u, 
then,  from  (53) 

dv8u  +  Sodu  ^GL-  EN    dvSv  ^  EM-FL 
duSa        ~FX-GM'    duU      FN-GM' 
from  which 

Eduhu  +  FydvZu  +  duhv)  +  Gdv^v  =  0,  (54) 

LduSu  4-  M{dv8u+  duho)  +  Ndvhv  =  0. 

From  the  first  of  these  equations  we  have,  by  (41),  the  following 
theorem : 

Theorem  I.  The  two  lines  of  curvature  at  a  point  on  the  sur- 
face are  orthogonal. 

From  the  second  equation  we  have,  by  (49),  the  further  theorem : 

Theorem  II.  The  tangents  to  the  lines  of  curvature  at  a  jioint 
on  the  surface  are  conjugate  directions. 

Conversely,  if  two  systems  of  curves  on  the  surface  are  orthog- 
onal and  conjugate,  their  equations  satisfy  (53)  and  (54),  hence 
they  are  lines  of  curvature. 

The  normals  to  the  surface  at  the  points  of  a  given  curve  O  on 
it  generate  a  ruled  surface.  The  ruled  surface  is  said  to  be  de- 
velopable if  the  limit  of  the  ratio  of  the  distance  between  the 
normals  to  two  points  P,  P'  on  C  to  the  arc  PP'  approaches  zero 
as  P'  approaches  P. 

It  should  be  noticed  that  in  particular  a  cone  satisfies  the  con- 
dition of  being  a  developable  surface.  A  cylinder  is  regarded  as 
a  limiting  case  of  a  cone,  and  is  included  among  developable 
surfaces. 

Theorem  III.  The  condition  that  the  normals  to  a  surface  at  the 
points  of  a  curve  on  it  describe  a  developable  is  that  the  curve  is  a 
line  of  curvature. 

Let  P  =  (it-,  y,  z)  and  P'  =  {x-\-  Ao;,  y  -f  Ay,  z  +  Az)  be  two 
points  on  the  given  curve  C.  The  equations  of  the  normals  at 
P  and  P  are  (Art.  20) 

X=a;  +  A.r,      Y=y-\-'fxr,     Z  —  z  +  vr, 
X  =  x  -I-  Ax  +  (A  +  AA)r',      Y  =  y  +  Ay  +  (jx  +  fxA)r', 
Z  =  z  +  Az  +(v  +  Av)r'. 


Abt.  206]  LINES  OF  CURVATURE  265 

The  ratio  of  the  distance  Al  to  the  arc  As  is  (Art.  23) 

Ai      Aic(/ZAv  —  vAjii)  +  A!/(vA\  —  AAv)  -|-  Az(XAJl—  /XAA) 

^^      ^s  ^QiAv  -  vA/X)2  ^  ^-^X  -  XAP)2  +  (AA/i  -  ;iAA)2* 

Divide  numerator  and  denominator  of  the  second  member  of  this 
equation  by  As^  and  pass  to  the  limit  as  As  =  0.  Using  the  dif- 
ferential notation  to  indicate  lim  Ax :  As,  etc.,  we  have 

|-      A^  _  dx(jidv  —  nljx)  +  dy(vdX  —  Xdv)  +  dz(\djl  —  fxdk) 

A*^  AS         V()idv  -  vdJxY  +  (vd\  -  Xd^y  +  (Ad/Z  -  Jxdxy 

Both  numerator  and  denominator  of  the  second  member  of  this 
equation  vanish  for  those  values  of  A,  jx,  v  which  satisfy  the 
equations 

dx     dfi dv     ■, 

A       ft        V 

and  the  limiting  value  of  the  ratio  -—  is  indeterminate.     The  de- 

As 

nominator  cannot  vanish  for  any  other  values  of  A,  'jx,  v. 

Since  A^  +  ^2  _^  v^  =  1, 

we  have,  by  differentiating, 

AcZA  -f  (xdji  +  vdv  =  0, 
which  reduces,  under  the  condition  that  dx  =  kX,  etc.,  to 
k{X'  +  H-'  +  P)=k=0. 

Since  k  —  0,  we  have  dx  =  dH-  =  dv  =  0.  Hence  the  normal  to 
the  surface  has  a  constant  direction  for  all  points  of  the  curve  C. 
The  surface  generated  by  the  normal  is  in  this  case  a  cylinder. 

If  the  denominator  of  (55)  is  not  zero,  the  condition  that  the 
surface  generated  by  normals  to  the  surface  along  C  is  a  develop- 
able is  that  the  numerator  of  the  second  member  of  (55)  is  zero, 
that  is,  that 

dx  Qldv  -  vdji)  +  dyCvdx  -  Xdv)  +  dz  (xdJL  -  JldX)  =  0. 

If  we  substitute  for  a,  Jjl,  v  their  values  from  (36)  and  for  dx,  dy, 

dz  their  values  --  du  -\-  -^-dv,  etc.,  we  can  reduce  this  equation  to 
au  dv 

(53),  which  proves  the  theorem. 


266  DIFFERENTIAL   GEOMETRY  [Chap.  XIV. 

207.  The  indicatrix.  Let  the  lines  of  curvature  be  chosen  for 
parametric  curves.  In  (54),  dv  =  0  and  8u  =  0,  but  du  =^  0,  Bv  ^  0, 
hence  F=0,  M=0. 

Let  C"  be  a  curve  making  an  angle  6  with  u  =  cons,  and  let  B 
be  the  radius  of  normal  curvature  in  the  direction  of  C".  Along 
u  =  cons.,  ds  =  ^Gdv,  hence  from  (41), 

cos  e  =  V^-,     sin  e=^E^. 
ds  ds 

From  (45)  and  (52)  we  now  have  the  formula 

1  _  cos^^      sin^^ 

This  equation  is  known  as  Euler's  formula  for  the  radius  of 
curvature  of  normal  sections.  It  is  intimately  connected  with  the 
shape  of  the  surface  about  P. 

Let  the  surface  be  referred  to  the  tangents  of  principal  curva- 
ture and  normal  at  P  as  X,  Y,  Z  axes. 

Let  X,  y  be  taken  as  parameters.  The  equation  in  x,  y,  z  has 
the  form 


Since  z  =  0  is  the  equation  of  the  tangent  plane  at  the  origin, 

1  —  1  =  0  and  I  —  1=  0.     Since  the  X  and  Faxes  are  the  tangents 
\dx)       _  \dyj  ^ 

of  principal  curvature  at  the  origin. 


,dxy      R^     \dxdyj         '     \dyy      Rn 

hence,  neglecting  powers  of  x  and  y  higher  than  the  second,  the 
approximate  equation  of  the  surface  for  points  near  (0,  0,  0)  is 

2;s=  — +  ^.  (56) 

Ri      R2 

If  —  and  —  are  both  different  from  zero,  the  surface  defined  by 
Ri  R2 

(56)  is  a  paraboloid.     If  one  of  them  is  zero  and  the  other  finite, 

the  surface  is  a  parabolic  cylinder.     If  both  are  zero,  the  surface 

is  the  tangent  plane  to  the  given  surface.     This  last  case  will  not 

be  considered  further. 


Art.  207]  THE   INDICATRIX  267 

The  section  of  the  quadric  (56)  by  a  plane  z  —  cons,  is  called  the 
indicatrix  of  the  given  surface  at  a  point  P. 

If  ^1  and  i?2  have  the  same  sign,  the  section  is  an  ellipse  for  a 
plane  on  one  side  of  the  tangent  plane,  and  is  imaginary  for  a 
plane  on  the  other  side.  In  the  neighborhood  of  P  the  surface 
lies  entirely  on  one  side  of  the  tangent  plane.  Such  a  point  P  is 
called  an  elliptic  point  on  the  surface. 

If  i?i  and  jRj  have  opposite  signs,  the  paraboloid  (56)  is  hyper- 
bolic and  the  section  by  any  plane  z  =  cons,  on  either  side  of  the 
tangent  plane  is  a  real  hyperbola.  The  point  P  is  in  this  case 
called  a  hyperbolic  point  on  the  surface. 

If  —  or  —  is  zero,  the  section  z  =  cons,  consists  of  two  paral- 
Pi  P2 
lei  lines  for  a  plane  on  one  side  of  the  tangent  plane,  and  is  im- 
aginary for  a  plane  on  the  other  side.  It  follows  from  (52)  that  at 
such  points  LN—  M^  =  0,  and  from  (47)  that  the  two  asymptotic 
tangents  coincide.  The  point  P  is  in  this  case  called  a  parabolic 
point  on  the  surface. 

In  all  three  cases,  the  directions  of  the  asymptotic  tangents  to 
the  surface  at  a  point  P  are  the  directions  of  the  asymptotes  of 
the  indicatrix.  At  an  elliptic  point  the  asymptotic  tangents  are 
imaginary;  at  a  hyperbolic  point  they  are  real  and  distinct;  at  a 
parabolic  point  they  are  coincident.  Moreover,  conjugate  tangents 
on  the  surface  are  parallel  to  conjugate  diameters  on  the  indica- 
trix.    The  asymptotic  tangents  are  self-conjugate. 

EXERCISES 

1.  Find  the  equation  of  the  tangent  plane  and  the  direction  cosines  of  the 
normal  to  the  surface  x  =  u  cos  v,  y  =  u  sin  v,  z  =  tfi  at  the  point  (m,  v). 

2.  Determine  the  differential  equation  of  the  asymptotic  lines  on  the  sur- 
face defined  in  Ex  1. 

3.  Show  that  the  parametric  curves  in  Ex.  1  are  orthogonal. 

4.  Find  the  lines  of  curvature  on  the  surface  x  =  a{u  +  u),  ?/  =  &(m  —  v), 
z  =  iiv. 

5.  Prove  that  if  E :  F  -.  G  =  L  .  M:  X  iov  every  point  of  a  surface,  the 
surface  is  either  a  sphere  or  a  plane. 


-  ANSWERS 

Page  3.     Art.  1 

I      2.   The  FZ-plane.  3.    The  Z-axis, 

'     4.   A  line  parallel  to  the  Z-axis  through  (a,  6,  0). 

6.    (k,  I,  —  to),  (k,  —  I,  in),  (k,  —  I,  —  m),  (—k,  I,  — m),  (  —  k,  —I,  — m). 

Page  5.     Art.  2 


3.    (-1,1,9).         4.    13.         5.    Vo^T&M^. 

6    —     A.     0-   —     —     — •    ^^     -^        2 

VS'    Vs'      '    Va'    V3'    VS'    \/89'    V89'    \/89 

0    J-     A.-       1         -4         2 

'  Vs'  Vs'   V21'   \/2l'   V2I' 

y  y 


y/x^  +  y^  +2^     y/x^  +  y'^  +  z^     Vx^  +  y'^  +  z^ 
Page  7.     Art.  4 


1.   V89.      4.    \/(x-l)2+(y-l)2+(2-l)2=V(x-2)H(2/-3)2+(2-4)2. 
A     r«^      2  3  5  ,,.111 

V38      V38      V38  VS      \/3      V3 

(c)  4=,  -4=,  ^. 

\/41      V41      \/41 
7.    (a)  Parallel  to  the  TZ-plane.         (6)  Parallel  to  the  Z-axis. 

(c)  Parallel  to  the  X-axis.         9.    ~,    -4,  —  • 

V3      \/3      V3 

Page  9.     Art.  6 

1.     :^.         2.    1.0,0;0,1,0;0,0,1.  3.    -4r,    ^^,    -i- 

1*  V2(j      V^      v26 


^^    •        8.   ^•s.  9.  2^         10    Twn      -uvV3-l, 


8.  (f,  I,  2).       10.  Two.     4-  V  V 3  -  1  •  11.(2,2,2), 


Page  11.     Art    9 

1.  Sphere  of  radius  1,  center  at  origin. 

2.  Cone  of  revolution,  with  X-axis  for  axis. 

269 


270  ANSWERS 

3.  Plane  through  Z-axis,  niakiug  angle  of  30^  with  JT-axis. 

4.  Cone  of  revolution,  with  Z-axis  for  axis. 

5.  (a)  p  =  2  ;     (6)  p  =  2  ;    (c)  p'^  +  z^  =  4.  6.    <p  =  45^  p2  =  g2, 

7.     Vpi^  4-  p2-  —  2  pip2(C0S  «!  cos  «o  +  COS  ^1  COS  ^2  +  cos  7i  COS  72)- 

Page  14.     Art.  12 

1.   3z  +  iy+2s=n.        2.    X  —  y  =  0.     X-  and  Y-  intercepts  zero. 
3.    -4x  +  3y  +  z^5.        i.   k  =  2.        5.    (-3,4,5). 

Page  18.     Art.  16 

1-   A  ^  -  if  .V  -  t\  2  =  2.         2.  x  +  2y  =  0. 

3.    -^-  4.    -4r-  5-    (4,  3,  1);   (1,  -4,  3). 

V26  \/l4 

7.   25  X  +  39  y  +  8  2  -  43  =  0.  8.    5  a;  -  1/  -  2  2;  -  6  =  0. 

g_   Aix  +  ^ly  +  C^i  +  Ji  ^  _^  Aox  +  Boy  +  C2g  +  D2 . 

10.  U(x:^  +  y''  +  z'-)  =  {Sx+y-2z-  11)2. 

11.  2  X  -j^  -  s  +  3  +  3 v6  =  0.  13.   X  -  2  y  -  2  +  2  =  0  ;   -  2,  1,  2. 
14.    21x- 9^-22^  +  63  =  0.            15.   3  x  +  2  y  +  3  s  -  15  =  0. 

17.  11  X  -  y  +  16  2  -  63  =  0  and  17  x  -  13  y  +  12  5;  -  63  =  0. 

18.  m  =  ±  6.  19.    k=-h 


1. 


Page  21.     Art.   20 

^   ^    \,   '    4      2/      \14  14  J      U     8       ; 


0). 


,x-3      2/-7      2-3         4  2         -3  ,         ,0,0      n 

4.    =  ^ = ■;    — ^^,    — :^  ,    — ^-  5.    x  +  2?/  +  2  =  0. 

4  2  -3        V29      \/29      v'29 

6.   k  =  ^.  7.   Yes.  8.   A;  =-2.  9.    No.  11.   Yes. 

Page  23.     Art.  21 
1     ^ni     _i_        5 
SVs'   3V3'   3V3 

2.  2x  +  2/  — 3^  +  6=0,     X +  2/  + 2 -13  =  0. 

3.  arc  sin     J^^  _ .  4.    x  +  10  y  +  7  2  +  18  =  0. 

V29  \/70 


ANSWERS 


271 


5. 

8. 

10. 
12. 

1. 
5. 


S  X  +  y  -  26  z  +  6  =  0. 

Sx-y  +  3z  -7  =  0. 

3  .r  -  7  2/  -  4  z  =  0. 
k  =  2  and  k  =  3. 


6.   k=-  1. 
9 


7.   X  4-  2  2  =  6. 


X  —  a    y  —  b    z  —  c\ 
li         mi         111     \—  0- 
h         mo         W2     I 


11.    k  =  l 

13.    2x-z  =  0,  y  =  3. 


Page  25.     Art.  23 
2.    \/3.  3.    0. 


4.    -4=,  0,   ^1,. 
VI 13  VllS 


V2;    X,    J,,   0. 

V2      V2 

Vl4. 
2 


6. 


1 

Vs' 


7.    15  cc  4- 43  =  0,    12  y  =  62 +13. 


Xl  —  y2      ^1       h 
2/1  —  2/2     WM     m2 

2l  —  22        «1        "2 


=  0. 


Page  28.     Art.  24 

61  x  -  52  2/  +  35  2  -  93  =  0.  3.    a;  +  5  y  -  3  0  -  44  =  0. 

12.T- 17  2/ +  32  +  4  =  0.  4.    Yes. 

7a;  +  12  2/-13  2  +  8  =  0,     x-3y  +  42-7  =  0. 

Page  29.     Art.  25 

7?/- 10  2-3  =  0,     7x-2-22  =  0,     I0x-y-3l=0. 

2/-2  +  2  =  0,     x  +  2  =  l,     x  +  2/=-l. 

2/  — 2=0,     x  +  22  =  4,     x  +  2y  =  i. 

(A1B2  -  AiBx)y  +  {AxC2  -  AtGi)z  +  {AiD-z  -  .i.A)  =  0. 

{B1A2  -  B2Ai)x  +  (5iC2  -  -B2Ci)2  +  (B1D2  -  B'Di)  =  0. 

(Ci^2  -  a2^i)x  +  ( C1B2  -  C2Bi)y  +  ( C1D2  -  C2D1)  =  0. 


1. 
2.    5. 

4.   arc  cos 

5. 
6. 

8.   A  plane 


Page  33.     Art.  28 

y 
2     4 


x  +  22/  +  J  +  l=0;    3x-|-|+l=0; 


x  +  |-|+l=0. 


3\/T0 


'•^'•^•-)^a-^-^)^(-i-'°) 


V299 

M  +  r  +  u>=-l,  6  7i-3»  +  tr  +  3  =  0,  6M-2»  +  ir+l=0. 

-1 


(2,  -1,  -3);   (— ,  0,  — 


0,  0, 


7.    —     — ,   — . 

V3'    VS'    V3 


9.    4(m'^  +  v2  +  w-)=  1.     A  sphere. 


^» 


272 


ANSWERS 


Page  35.     Art.  34 

(c)(^,o,o).  (/)(«.  ".n)- 

2.    (-10,15,-2,0).  4.   7x  +  9j/  +  542-59«  =  0. 

Page  37.     Art.  35 

1.    (a)  Parallel  bundle.     Rank  3.         (b)  Rank  4.  (c)  Rank  4. 


(d)  Parallel  bundle.     Rank  3. 

-1         c  b 


3.    The  determinant 
-69  5 


4. 


a  —  1    c 
6    a  -  1 
-  19 


is  of  rank  3 ;  of  rank  2 ;  of  rank  1. 


\/3867   \/3867   \/3867 


Page  43.     Art.  40 

1,   x2-3  2/2  +  2/2-4x-8y  +  42  +  4  =  0. 

^21      V6      VU 

^ix'        y'    ^       z' 

V21      V6      vTi' 

z  ^  ^x'        y'        Zz' 

~  V21      V6      Vli 

4.   New  equation  is  x^  —  2  j/^  +  6  2^  =  49. 

Translation  is  x  =  x'  +  8,  ?/  =  y'  —  1,  2  =  2'  +  2. 
6.   3  x2  +  6  y2  +  18  ^2  =  12. 

Page  45.    Art.  41 

-     /28±6i      -6±8i     5T24i\  ^      /17-4t 


)    -  (^- H^' -*'•)■ 


13      '        13      '         13      ;  "     V       6 

6.    (13  +  9i)x+(3  +  4i)2/  +  (16-7i>  =  23  +  64  i. 

6.    (1  ±  iV-T,  0,  0). 

Page  46.     Art.  42 

1.   x2  +  j/2  =  4  z^.  3.   x2  +  y'^  +  2^  -  7  X  +  .V  +  30  =  0. 

4.    8(x2  +  y2  ^.  ^2)  _  68  X  +  48  2/  -  66  2  +  275  =  0. 


ANSWERS  273 

5.  «  =  5,  (x-3)2+(2/-7)2  +  (;j-l)2  =  9. 

6.  2  a;  -  14  y  -  2  3  +  1  =  0,  4  X  -  18  0  +  33  =  0.  7.    (-  4,  ±  4  i,  2). 

Page  49.     Art.  46 
8.   Center  at  (0,0,  4);  radius  6.        4.   x"^  -  y"^  =  1.        6.   r,  — • 

Page  51.     Art.  47 

1.   a;2  +  2/2  +  ^2  =  26.        2.  9(y2  +  z^)  =  (15  -  2  a;)*.     Vertex  (Y,  0,  0) ; 
z  =  0,  9(2/2  +  ^2)  =  225  ;  4(x2  +  z^)  =  9(5  -  y)^. 

4.    2/2  +  22  =  (,2  ;    y  =  a. 

6.     (a)?!  +  l^4-^=l;    ?-%  1^  +  ^' =  1. 
^^499  '     494 

^  '  a2      62      fc2         '    a2      1-1      a2 

(c)  2/2  +  22  =:  8  x ;  2/*  =  64(x2  +  z^). 

(d)  (x2  +  2/2  +  ^2  _  5)2  =  16  -  4  x2 ;  x2  +  (y  -  1)2  +  22  =  4. 


(e)  2/2  +  ■^2  =  sin2  x ;  ?/  =  sin  V'x2  4-  22. 
(/)  y2  +  22  =  e2'  ;  2/  =  e-^x^'. 

Page  54.     Art.  49 

1.    (o)  x2  +  2/2  +  z2  :=  ^2.  (6)   x2  +  2/2  +  22  +  2  x  -  8  y  -  4  2  =  16. 

(c)  x2  +  y2  +  22  -  4  x  -  2  y  -  10  2  +  14  =  0. 


(a)  Center  f- I,  -1,  --\  radius  ^^. 
^   '  V      2  '       2/  2 

(6)  Center  (-1,  -  2,  3) ;  radius  0. 

(c)  Center  fl,  1,   ^V  radius  ^' ^^^ 
^  ^  U    2       4    r 


lius 
2 


4 


(d)  Center  (  _  /,  0,  O^j;  radius 

8.    (-4±3i,  2±6i,  5,  0). 

.     /2±2iV2      1  T  2  t  V2      _2±\/2i\ 
I         3  '  3         '  3  } 


Page  56.     Art.  52 

1.  y  =  1. 

2.  Arc  cos .     The  spheres  have  no  real  point  in  common. 

8.   x2  +  y2  +  22  _  2  X  -  6  y  -  6  2  +  10  =  0      and      x2  +  y2  +  22  -  2  x  -  6  y 
-62-6=0, 


274  ANSWERS 

4.  x^  +  y^  +  z^  +  x-2y-Sz  =  0. 

5.  2  X  —  o  y  +  z  +  5  =  0.    The  sphere  is  composite. 

6.  10(x2  +  y2  +  2-2)  ^-jix  -6Sy  -8[)z  -  185  =  0. 

7.  4232(a;'-!  +  y-^  +  z^)  -  276  a;  +  27G  y  +  1032  ^  +  225  =  0. 

Page  69.     Art.  59 

1.   Center  (1,  1,  -2);  semi-axes^,  — ,  — • 
^  '    '         ^'  2  3         2 


2.    Sphere  ;  center  [ 2,  '-,    —  5j;  radii 


V'2U5 


2 

3.  y  =  0,  2  x^  =  3  z^  +  5  z  +  7 .     Rotated  about  the  Z-axis. 

4.  X  =  1,  y  =  z ;   x  =  1,  y  =—  z;   x  —  —  1,  y  =  z;   x  =  —  I,  y  =—  z. 

5.  (a)  Ellipsoid,  (ft)  Hyperboloid  of  two  sheets,  (c)  Hyperboloid  of 
one  sheet.  (cZ)  Hyperboloid  of  revolution  of  one  sheet,  (e)  Ellipsoid. 
(/)  Imaginary  ellipsoid. 

Page  73.     Art.  64 

1.    Hyperboloid  of  one  sheet.  2.    Imaginary  cylinder. 

3.    Elliptic  paraboloid.        4.    Real  cone.        5.    Hyperboloid  of  two  sheets. 

6.  Hyperbolic  paraboloid.  7.   —  +  ^   "*"  ^  =  1. 

8.    (a)   (1  -  r-2)a;2  +  y^  +  z-i  -  2  ax  +  d^  =  0. 

(b)  (1  -  r2)x2  +  (1  -  r2)j/2  +  z^-2ax  +  a^  =  0. 

Page  76.     Art.  66 

^      /-8±n/I09     5  T  V109      -  17  ±  VlO!)\  g     fO    0    0^ 

'    \  3  '  3         '  (J  j  •    V  .     7     > 

3.    (-1,1,   -A)-  4.    (-1,  2,-1).  5.(1,1,0). 

6.   Vertex  (0,  -1,0).  7.    Plane  of  centers  2(x  -  y +  «)- 1  =  0. 

8.    Non-central. 

Page  89.     Art.  75 

1.  Hyperboloid  of  two  .sheets.     Center  (0,  0,  0).     Direction  cosines  of  axes 

2      _    2      1.2      1      2.1      2      2  fl  r2  -U  9  »;2 .^i  4_  2  —  0 

2.  Hyperboloid  of  one  sheet.    Center  (1,  |,  —  f).    Direction  cosines  of  axes 
2  +  2-\/5  VS-l  5+\/5       .       2-2V5  -\/5  -1 


2V15  +  4V6     2Vl5+4v'5     2Vl5  +  4\/5     2Vl5-4\/5     2Vl5-4V5 
5_V5       .     -3         4  2  5  +  V5^2   ,   •''>  -  v^^' ^2  _  3  ^2  =  10 


2V15-4V5'    ^29      V29      v/29  22  3 


cosines 


ANSWERS  275 

3.  Hyperboloid  of  one  sheet.     Center  (  — ,  -pr- ,  -pr-  1  • 

Direction  cosines  of  axes,  7n'^'^'^'^';ll';;|5' 
^'^-     11-^  +  4  2/^-^^  =  A. 

2        1 

4.  Real  cone.     Vertex,  (1,  0,  0).      Direction  cosines  of  axes,  -7=  >  — 7=» 
-12         11-25  V5    V5 

Ve   Ve   a/6    V30   V30    V30  ^ 

6.  Elliptic  paraboloid.      New  origin,  (  -  ,   —  ,   —  1 .    Direction  ^ 

2        1       -1.1       -1        l.n-1-1 

of  axes,  — F )  -7=  >  —7= )  -7= )  —7=  >  —7=  >  ">  -7=  >  —7=  •     6  x2  +  3  2/2  = 

Ve   Ve    Ve    V3    V3    V3        V2   V2  -^     y 

(4     - 1    22  \ 
^  '   -7-  >   57  1 .     Direction 

1-5         3  1  2  3-3^1 

cosines  oi  axes,       / —  >  ~7= »      , —  >      , —  >      , —  »      , — :  >      , —  >   ">      , —  • 
V35     V35     V35     Vl4     Vl4     Vl4     VlO  VlO 

2  x2  -  5  2/2  +  7  z2  +  —   =0. 

35 

7.  Parabolic  cylinder.     New  origin  on2x  +  22/-2z-l=0.    2x  -  y 

2—1222      2        — 1 

+  2  z  -  2  =  0.     Direction  cosines  of  axes,  -  >    -r-  >    -  j    ^  >    ^  >    ^  J      — -  , 

cy         c}  *j  O  *J  O         fj  (j  O 

3  >    -  •     5  2/2  +  6  X  =  0. 

8.  Two  real  intersecting  planes.       Line  of  vertices,    x  +  z  -  1   =  0, 

X  -  2/  +  2  =  0.     Direction  cosines  of  axes, —7^1  ^^  >  —7=^;  0,  ~7^  >  —p^', 
_1     _1       1         021,  'V6     V6     V6'     '  \/2     V2' 

ZJ:^    ij   _L_.      3x^       l_y2 

V3'  Vs'  ^3'      2    ~    2     " 

9.  Hyperbolic  paraboloid.    New  origin,  (1,  0,  -  1).    Direction  cosines 

of     axes,     -^ ,    _?^ ,    _L  ;     J_  ,    III  ,    J_  ;    ^  ,0,     —  .     fi   j-2    _    q    >/2    - 

2^  z.    ^6'  Ve'  Ve'  V3'  V3'  V3    V2'       V2    ^  "^      ^  ^  - 

10.  Elliptic  paraboloid.     New  origin,  (0,  -  1,  1).    Direction  cosines  of 

1  1  1.1-1^1  1-2 

axes,   -7= '   -7= ,   -7= ,    —^  ,—=.,   U  ;-—,—=,-—=  .     3  x2  +  4  2/^  = 
V3     V3     V3      V2      V2  V6      Ve      V6  -r       « 

sVe  z. 

11.  Hyperbolic  cylinder.      Line  of  centers  3x-72/  +  7z  +  l=0,  or 

2x-22/  +  4z  +  l=0.  7X  +  52/  + 112  +  5  =  0. 

X  +  32/+     2  +  1=0. 

1—12  1  3  1  — 7 

Direction  cosines  of  axes,   —7=  >   -7^ .   — ;=^ !    — f=  »   —7^:^  .    —7=^  J    -7=  , 

14  '  Ve    Ve    Ve     \/n    Vu    Vn     Vee 

Vl'    V^-    24x2-112/2-5  =  0. 


276  ANSWERS 

12.  Hyperbolic  paraboloid.     Origin  (  —  ,    —  ,    ^—  )•      Direction  cosines 

\72      72       i2  I 

f  1  V7-2  V7-3  -1 

of  axes  —  — ,   • —  -  ,   —  - ;   —  — , 

V28-10\/7     V28-  lOvT     V28  -  10  V?      V28  +  10  V7 

V7  +  2  V7  +  3  1        -1       1 

V28+IOVT     V28+IOV7      ^     >/3      V3 
(_  1  +  V7)a;'^ -(1  +  ^7)2/2  =  4  V3^. 

13.  Hyperboloid  of  one  sheet.     Center  (^,  ^-  ,  — V      Direction  cosines 

of  axes  .21,  -  .65,  .69;  .91,  .41,  .10  ;  .36,  -  .64,  -  .68. 
3.09  a;2  +  1.59  y2  _  3.67  ^2  ^  as, 

(2     3    —  26\ 
— ,  - ,  ) .     Direction  cosines 
15    5      15    / 

of  axes   -.77,    .56,   .28;    .14,    -.31,    .94;   .63,  .76,  .13.         6.17  x2 +  . 712,2 
-6.8822=^^9. 

15.    Ellipsoid.     Center  (0,  1,  1).     Direction  cosines  of  axes 

2  1+^^      .0=  -^         .      -^+^L,0:0.o,i. 


V'l0+2V5      V1O  +  2V5  V1O-2V5      V1O-2V6 

— x^  -\ y^  +  2  z^  —  i. 

2  2 

(—9—7  \ 
,  ,  —  6  1.     Direction  cosines  of  axes  .83, 

-  .33,  -  .44 ;  .26,  .95,  -  .22  ;  .49,  .07,  .87.  4.20  a;2  +  .59  y^  +  .20  z^  =  ^. 

17.   I,  5±2i.  18.    ^=^. 

^'  28 

Page  92.     Art.  78 

1.    X+IO2/-32  +22  =  0,  ^^^zJ:  =  1+l  =  ?^zl. 
"  1  10-3 

Page  96.     Art.  80 

1.   y  -i,  z  =  kx,;  x--n,  z  =  -r)y. 

Page  97.     Art.  81 

1.    V5,  :^.         2.   a;  +  z  +  1  =  0,  2/  +  2  -  1  =  0.         3.   a,  6,  c. 


ANSWERS  277 

Page  103.     Art.  83 

1.   x  +  y  —  z  =  dandx- y +  2z  =  p.        2.  x  —  {2  ±  y/6)y  =  d. 

3.   -^.         4.    y  +  S±\/2{z-2)-0.  5.   a  =  6,  h=0. 

V3 

6.   ax  +  fir.2;  +  Z  =  0,  ay  +  fz  +  m  =  0.  7.    2  g'x  +  2/y  4-(c  -  a)^  =  d. 

8.  (6 -i-)^- +(«-^')-S2 -2/4^5  =  0, 
(c  -  i-)-S2  +  C?>  -  A-)  C2  -  2  /J5C'  =  0, 
(a  -  A;)  C2  +  (c  -  A)^^  _  2  ^C^  =  0, 

A"  being  a  root  of  the  discriminating  cubic. 

9.  (-1,0,  -3).     ^. 

Page  108.     Art.  87 

8.  ki  =  cons,  i  =  1,  2,  3,  4.  For  parametric  equations,  substitute  this  value 
of*,-,  inEqs.  (27). 

Page  111.     Art.  89 

1.  (-6x4-61/- 12 «,  x  +  2y-2z  +  t,  Qx  +  Sy  +  iz  +  it,  —x  +  Sy 
-z-2t),  (-12,  1,  4,  -2),  (12,  -2,  -20,  1),  (18,  -6,  -16,  1), 
(12,-3,  -28,1),    (3,2,1,2). 

2.  (-373,  179,  92,  283),  (-500,  181,  145,  344),  (-153,  61,  38,  107), 
(-37x1-96x2-9x3  +  156x4,      11  Xi  +  24  X2  -  3  X3  -  60  X4,      8x1 +  48x2 

-  6  xa  -  36  X4,   31  Xi  +  60  X2  +  3  X3  -  108  X4). 

3.  15  X  +  5  J/ +  11  0  +  16  e  =  0. 

4.  197  xi  +  468  X2  +  57  X3  -  792  X4  =  0. 

5.  6x'^-15y^  +  2  z^  +  3  yz-  zx- 3  xy  + n  xt  + 9  yt- 6  zt  + 10  t^  =  0. 

6.  (22  X  -  22  2/  +  44  «,  12  x  +  24  j/  -  24  0  +  12  «,  33  x  +  33  y  +  22  2;  +  22 1, 
66  x-198  2/ +  66^+132  0,  (22,  6,  11,  66),  (22,  12,  55,  33),  (33,  36,  44,  33), 
(22,  18,  77,  33),  (22,  -48,  -11,  264),  (-97121,  36427,  22804,  66851), 
(296167,  -  115487,  -  64346,  -  205981),  (-  185625,  71181,  42570,  128403), 
(814  xi  -  6912  X2  -  297  X3  -  61776  X4,     -  242  xi  +  1728  X2  -  99  X3  +  23760  X4, 

-  176  xi  +  3456  X2  -  198  X3  +  14256  X4,     -682  Xi+4320  X2+99  X3  +  42768  X4). 

Page  113.     Art.  92 

1.  Vertices :  ?(i  =  0,  (1,  0,  0,  0) ;   M2  =  0,  (0,  1,  0,  0) ; 

M3  =  0,  (0,  0,  1,  0);   Ui  =  0,  (0,  0,  0,  1). 

Faces  :        Xi  =  0,  (1,  0,  0,  0);   X2  =  0,  (0,  1,  0,  0); 

X3  =  0,  (0,  0,  1,  0);   X4  =  0,  (0,  0,  0,  1). 

2.  Xi  =  0,  X2  =  0  ;  M3  =  0,  M4  =  0. 
Xi  =  0,  X3  =  0  ;  ?<2  =  0,  M4  =  0. 
iCi  =  0,   X4  r=  0 ;    U2  —  0,  W3  =  0. 


278  ANSWERS 

X2  =  0,  Xa  =  0 ;  Ml  =  0,  Hi  =  0. 
X2  =  0,  a;4  =  0 ;  Ml  =  0,  u^  =  0. 
Xs  =  0,  a;4  =  0 ;   Ml  =  0,  M2  =  0. 

3.  Ill  +  ^'2  +  ^(3  +  W4  =  0,    3  Ml  —  5  M2  +  7  Ms  —  W4  =  0, 

—  Ml  +  6  2t2  —  4  M3  +  2  M4  =  0,  7  Ml  +  2  M2  +  4  M3  +  6  M4  =  0. 

4.  (1,  1,  1,  1),   (7,  -1,-3,  1),   (1,  9,  -  6,  2). 

5.  Ml  -  M2  =  0,    7  Ms  +  U4  =  0.  6.    (-  9,  1,  1,  0). 

Page  117.     Art.  95 

2.  pxi  =  li  + 2  12  +  10  h, 

px2  =  lh  +  5h-  h,  (176,  -  175,  40,  363). 

pX3=—h  +  4:l2-S  h, 

px4  =  3li  +  h—  ^h. 

3.  pui  -—bh  +  lh  +  Gh, 

pu2  =  3h-5h-^h,  (21,32,1,5). 

pMs  =  4  Zi  +  3  ^2  -  3  Zs, 

J9M4  =  ?1  +  2  Z2  +  h- 

5.  ^Mi  =  Zi  +  7  I2,  pu2  =  —  5  Zi  +  2  ^2,  puz  =  3  Zi  —  Z2,   pUi  =—  h  —  h. 

6.  px\  =  Zi  +  3  Z2,    pa;2  =  2  Zi  —  2  Z2,    px^  =  —  3  Zi  +  5  Z2,    pXi=—h-2  h. 

Zi(Mi  +  2  M2  —  3  Ms  —  U4)  +  Z2(3  Ml  —  2  ?f2  +  6  7<3  —  2  M4)  =  0. 

Page  120.     Art.  97 

3.  («11  +  «12  +  «13  +  «14,  "21  +  «22  +  «2S  +  «24,  Csi  +  «32  +  "33  +  a34, 
a41  +  ^42  +  «43  +  «44).  (/3ll  +  )321  +  ftl  +  i34l,  j3l2  +  1822  +  /332  +  /342, 
/3l3  +  ^23  +  ft3  +  ^43,     /3l4  +  ^24  +  ^4  +  /344)- 

4.  xi  =  A;iXi',   X2  =  kiXi',   Xs  =  k^x^',   X4  =  k4X4'. 

Page  122.     Art.  100 

1.    (a)  xi  =  Xi' —  3:4',   a-2  =  X2' —  X4',   Xs  =  xs' —  X4',    X4=— X4'. 
D(p)  =  (l  +  p)(l  —  py.     Invariant  points  are   (1,  1,  1,  2)  and  all  the 
points  of  X4  =  0. 

(6)Xi  =  X2',     X2=Xi',     X3  =  X4',     X4  =  Xs'.       D(p)  =  (p^-iy. 

Every  point  on  each  of  the  lines 

Xi  +  X2  =  0,  Xs  +  X4  =  0 ;   xi  —  X2  =  0,  X3  —  X4  =  0. 

(C)    Xi=X3',    X2  =  Xl',    X3  =  X2',    X4  =  X4'.      D{p)  =  {I  —  p^ip"^  +  p  +  1). 

The  points  (1,  w,  w^,  0),  (1,  u"^,  w,  0),  w^  _  j^  and  every  point  of  the  line 

Xl  =  X2  =  Xs. 

(d)  Xl  =  —  X4',   X2  =  Xl'  —  X4',   Xs  =  X2'  —  X4',   X4  =  Xs'  —  X4'. 

D(p)=p*+p^  +  p''+p  +  i.  (d,  i  +  0,  -e»(i  +  e),  -6^),  6^  =  1,  e^i. 

3.    "'*  =  cons.     i,k  =  l,  2,  3,  4. 


ANSWERS  279 

4.    In  case  X3  =  :cs\  the  point  (0,  0,  0,  1)  and  all  the  points  of  the  plane 

3-4=0. 

In  case  Xs  =  —  xz',  every  point  of  each  of  the  lines  xi  =  0,  a-2  =  0 ;  3-3  =  0, 
Xi  =  0. 

6.  (1,  1,  1,  1),  (1,  -  1,  1,  -  1),   (1,  i,  -1,   _  I),  (1,  - 1,   -  1,  i). 

7.  All  the  points  in  the  plane  at  infinity. 

9.    ^'^. 

Page  125.     Art.  102 

1.  X3  —  aXi  =  0. 

3.  (2  ±  2  iy/l2f,   -  3  T  iVl2f,  8,  4). 

Page  131.     Art.  106 

2.  A=-  1. 

abed' 

6.  bcui^  +  cath'^  4-  2  abuzUi  =  0. 

7.  A  =  ^.     *(?«)  =  M2^  —  U1U2  +  U1U3  —  U2U3  —  iiiiu  +  M2W4  —  2  M3W4  =  0. 

8.  ^(x)=0. 

9.  Xi  =  0,  X3  —  3:4  =  0  and  Xi  =  0,  Xs  +  X4  =  0. 
10.    an  =0.     1=  1,  2,  3,  4. 

12.  a,fc2  =  a^i .  a^j,  /,  k  -  1,  2,  3,  4. 

13.  A  conic  ;  two  distinct  points  ;  two  coincident  points. 

Page  134.     Art.  Ill 

1.   xi  +  a;2  +  xs  +  a;4  =  0.  2.  X3  =  0,  X4  =  0. 

4.  (2  Xi  +  X2  -  3  X3  —  X4)2  +  4(XiX2  -  X3X4)  =  0. 

7.  XiSWiX,-  —  XiZvXi  =  0.     Three. 

8.  013X1X3  +  ai4XiX4  +  023X2X3  +  a24X.X4  =  0. 

Page  141.     Art.  118 

3.  aiixi^  +  022^^2-  +  «33^"3'^  +  2  ai2XiX2  +  2  023X2X3  +  2  013X1X3  =  0. 

4.  oiiXi^  +  022X2*  +  2  012X1X2  +  2  023a-2a;3  +  2  013X1X3  =  0. 

Xi  =0,    X2  =  0. 

Page  143.     Art.  120 

1.  8  xi2  +  xo2-5  X32-2  X42  +  9  X1X2  +  5  X1X3  + 18  X1X4  +  13  X2X4  -7  X3X4=0. 

2.  i(xi  +  X3)  -  (X2  +  X4)  =  0,        i(xi  +  X4)  -  (X2  -  X3)  =  0,       and 
i(xi  +  X3)  +  (X2  +  X4)  =  0,         iXxi  +  X4)  +  (a-2  -  X3)  =  0. 

5.  Equations  of  faces  20^4X4  =0,     i  =  1,  2,  3,  4. 


280 


ANSWERS 


Page  146.     Art.  122 

(0,0,2,  -3).  2.    k  =  ±4. 

72  Mi^  +  36  U2^  +  23  H3-  -  54  Mi?t2  =  0  ;  2  ?t3  -  3  «4  =  0. 

(a)  A  quartic  curve  with  double  point  at  0. 

(b)  A  cubic  curve  passing  through  0. 

(c)  A  plane  section  of  K,  not  passing  through  O. 


3. 


Page  150.     Art.  126 


2  \<^ik  —  ^^ifc 


an  —  Xbii 

(a)  X  -  1,  X2  X,   [1(21)]. 
(6)  M  [4]. 


«il 

«.3 

aA2 

aw 

"ml 

"mS 

aj2 

«!3 

(C)    X2,    X2 

((Z)   X3,   X- 
Art.  131 


Page  156. 

[111].   Four  distinct  lines. 

[21].    Two  distinct  and  two  coincident  lines. 

[1(11)].   Two  pairs  of  coincident  lines. 

[3].    Three  coincident  lines  and  one  distinct  line. 

[(21)].        Four  coincident  lines. 

[(111)].   A  quadric  cone. 

{3}.   A  plane  and  a  line. 

(a)  X-1,  X-J,  X-^ 

(6)  X  +  ^,  X  -  i,  X  -  i. 

(c)  (X-l)2,  X-^. 

(d)  (X-l)3. 

4.    (a)  Four  distinct  lines. 

(6)  Two  pairs  of  coincident  lines. 

(c)  Two  distinct  and  two  coincident  lines. 

(d)  Three  coincident  lines  and  one  distinct  line. 


[111]. 

[1(11)]- 

[21]. 

[3]. 


(«)  a:i-  +  ^  +  ^ -  X  (xi^  +  3:2^  +  X32) 


:0. 


(b) 


xr 


41 


+  '^'^^'  +  '"'«")  -  X(a;i2  +  X22  +  3:3^)  =  0. 


[(22)]. 
[31]. 


(c)  ^  +  2  X2X3  +  X2^  -  X(xi2  +  2  X2X3)  =  0. 

(d)  2  X1X2  +  X32  +  2  X2X3  -  X(2  X1X2  +  X32)  =  0. 

Page  164.     Art.  133 

2.    [11(11)].      VXi  -  X3X1  ±  VXa  -  X2X2  =  0,  Xi2  +  X22  +  X32  +  X42  =  0. 
[1(21)].       V\2  -  XiXi  ±  X3  =  0,  Xi2  +  2  a:2.r3  +  X42  =  0. 

[1(111)].       Xi=0,    X22  +  X32  +  X4'^  =  0. 


ANSWERS  281 

[22].  n  =0,  a-2  =  0. 

[2(11)].       Xi  +  ix3  =  0,  3:4=0  ;  xi— 1X2=0,  X4=0;  X3=0,  3ri2+a;2-  + 3-42=0. 

[(11)  (11)].  Xi  +  1X2  =  0,  Xs  +  1X4  =  0 ;  Xi  +  1X2  =  0,  X3  —  1x4  =  0  ; 
Xi  —  1x2  =  0,  X3  +  JX4  =  0  ;  xi  —  ix2  =  0,  Xg  —  1X4  =  0. 

[4].      X2  =  0,  X4  =  0. 

[(22)].        X2  =  0,  X3  =  0  ;  xi  =  0,  X4  =  0;  X3  =  0,  X4  =  0;  the  last  one 
counted  twice. 

[(31)].         xi  +  JX4  =  0,  X3  =  0;  xi— (>4=0,  X3=0  ;  X4=0,  Xi'-  + 2x2X3=0. 

[(211)].       Xi  =  0,  .r3  =  0  ;  Xi  =  0,  X4  =  0. 

[{3}1] .  X2  =  X3  =  0  ;  xi  -  nxi  =  0,  2  x^^Xi  +  X3-  =  0. 

3.  (a)   (X  -  i)2,  (\  -  ^y.     xi  -  X2  =  0,  xs  +  2  X4  =  0  ; 

Xi  -  X2  -  V3(X3  +  2  X4)=  0,    V^(xi  +  X2)  +  2  X3  +  Xi  =  0  ; 

Xl  —  X2  +  V3(X3  +  2X4)  =  0.    V  o(Xi  +  X2)—  2  X3  —  X4  =  0. 

X1X3  +  X2X4  +  2  X3X4  —  X(2  X1X3  +  2  X2X4)  =  0. 

(b)  X  —  1,  X  —  1,  (X  +  1)2.  xi  +  X3  =  0,  xi  —  X2  +  iXi  =  0  ;   Xi  +  X3  =  0, 
.71  —  X2  —  1x4  =  0 ; 

a:i  +  X3  +  4  X2  =  0,  x'-4  +  (xi  —  X2)2  —  24  X2'-  —  16  X2X3  =  0. 
xi^  +  X2^  +  X42  —  2  X3X4  —  X(xi2  +  X22  +  X42  +  2  X3X4)  =  0. 

(c)  X  +  3,   X  -  1,    X  -  1,  X  -  1.     xi  +  2  X3  +  X4  =  0,    5  xi2  -  xr  +  6  Xs* 
+  4  X1X3  +  2  X1X2  =  0. 

-  3  xi^  +  xr  +  X32  +  xr  —  X(xi2  +  X2-  +  X32  +  x^^)  -  0. 

(d)  X  —  1,      X  —  1,  X-.       xi  +  X2  =  0,     Xi  +  ,T3  +  X4  =  0  ;      xi  +  X2  =  0, 
xi  —  X3  —  X4  =  0  ;  X2  +  X4  =  0,  3  x\-  +  X22  —  X32  +  4  X1X2  +  2  X2X3  =  0. 

Xi'-  +  X22  +  X4'-  —  X(xi2  +  X22  +  X42  +  2  X3X4)  =  0. 

4.  [1(111)].     [2(11)]. 

Page  167.     Art.  135 

1.  (X3  +  2  X2  +  4  X  +  l)wi2  +  (3  X2  +  7  X  -  10)K2='  +  (X^'  +  2  X2  +  9  X  +Q)ui^ 

+  (X2-  l)(\_l)M42_6(X2_\)„i„2+   12(X^+   1)2hiH3  +  Q{\'^  -  l)uxU4 

+  4(X2-  x)?<2?<3  +  2 x(x  -  i)2m2M4  -  4(x2  -  i') mm  =  0. 

2.  2  X22  —  3  X42  +  6  X1X4  +  2  X2X4  —  4  X3X4  =  0,  twice. 
2  xi2  +  2  X32  +  3  X42  —  6  X1X4  +  2  X2X4  +  4  X3X4  =  0. 

3.  2(miM2  +  M3M4)^^  +  ("2^  —  6  a?«i«2  —  6  auzUi)X- 

+  (6  a?uiU2  +  6  a?UiU4  —  2  au2^)\  +  a2M22  —  2  ahtiti2  —  2  ahisUi  =  0, 

Page  174.     Art.  142 

3.  (a)   [211].         (5)   [22].         (c)   [31]. 

4.  All  the  quadrics  of  the  bundle  toucli  a  fixed  line  at  a  fixed  point. 

5.  The  quadrics  touch  x^  =  0,  X2  —  2  X4  =  0  at  (0,  2,  0,  1),  and  X3  =  0, 
X2  +  2x4  =  0  at  (0,  2,  0,  —1);  they  have  four  basis  points  in  the  plane 
X2  —  X3  =  0,  at  the  points 


282  ANSWERS 

(2,  2,  2,  V3),  (2,  2,  2,  -V>]),  (-2,  2,  2,  \/3),  (-2,  2,  2,  -VS). 
y'l  =  -1  yrii4,     y'o  =  -^  2/12/4  (^  2/3  -  ^2), 
2/'3  =  4  ?/i(/32/4,     /y'4  =  2/1  (-  2/;!''  +  '^  2/22/3  —  2/2^). 
7.    Xi(Ma;)(Mx')[(«"'x)(M"x")-(?t"x)(?("'x')] 
+  \2{u"x){u"x')\_{u'x){ux')-{nx){u'x')^ 
+  \zl{u'x){u"'x){ux'){u'ix')-{ux){%i"x){u'x'){ti"'x')'\^0. 
(For  notation,  see  Art.  119.) 

Page  180.     Art.  146 

2.   yvyiy^yi  =  0.        3.    The  plane  counted  twice  is  a  quadric  of  the  web. 

4.  (x^  +  y- +  Z')fi  =  0. 

6.    Any  point  on  x^  +  y'^  -{-  z^  =  0  is  conjugate  to  any  point  on  «  =  0. 

Page  187.     Art.  150 
1.    (SMix'i)^  =  0.         2.   8.         4.   5. 

5.  Xi(?t'-^  —  W-)  +  \2(i)-  —  r«-)  +  Xs^n;  +  \iVio  +  XjMj?j  =  0. 

6.  [1111].     (a2  _  c2)m-;  +  (/)•■!  _  c-^)v2  =  s'^ ; 

(a^  —  6-)?|2  _  (ft2  _  c2)^2  —  s2  ; 

(a^  -  62)  m2  +  (a2  _  c2)i,2  ^  s2  =  0  ; 

y2    -f  ■j;2  ^    ^(;2   _  0. 

Page  196.     Art.  155 

1.  kiP  =  kiP  =  JC3P  =  kiP. 

Page  205.     Art.  160 

2.  (x2  +  2/2  +  ^2)2  =  ^^^  +  g  +  g^f2.       Eigllt. 

3.  2  0(x2  +  y-  +  Z-)  =  (ax-  +  by-)t.     Eight.     Fifteen. 

7.  xi  =  .r2'(.)-i'  +  .r4')(-^2'  +  :*-4')'         ^2  =  a;2'a-4'(xi'  +  X4'), 

X3  =  X2'Xi'{X2'  +  Xi'),  X4  —  Xz'Xi'{Xi'  +  ^i')- 

(1,  0,  0,  0),   (0,1,0,0)  ;  the  line  xi  =  0,  a-o  =  0.     Touch  at  (0,  0,  0,  1). 

8.  Xi  =  Xi'Xs'iXl'Xi'  +  Xi'Xs'  +  Xs'Xl'),        X2  =  X\'X2'{Xi'X2'  +  X2'X3'  +  Xz'Xx'), 
Xz  =  X2'X3'{Xi'X2'  +  X^'Xz'   +  X^'Xl'),        X4  -  Xi'xJ Xz'Xi' . 

(1,  0,  0,  0),   (0,  1,  0,  0),   (1,  0,  0,  0).     Four  coincident  at  (0,  0,  0,  1). 

Page  207.     Art.  162 

1.    A„/(a;)  =  4  2/i(«40ooa;i-'  +  3  02200X112-  +  3  a202oa;i«3^  +  3  a20023'iX42) 

+  4  2/2(3  ffl2200'>'l'^-'*"2  +  «0400a'2^  +  3  ao22oa;2*'3"  +  3  ao202X2X42) 
+  4  2/3(3  a2020''l23-3  +  3  nQ22QXrXZ  +  n'0040^3'  +  3  rt00225C3^4^) 
+  4  J/4  (3  a2002.''l''.r4  +  3  ao202a'2'^3'4  +  3  ao022X32.r4  +  ffl0O04a;4^)  • 


ANSWERS  283 

^U^fip^)  =  12  ?/l-(«4000-'>'l-  +  «2200^'2-  +  «2020a'3"^  +  «2002a^4-) 
+  12  y2'-^(ffl2200a;i'-^  +  (?0400-^2-  +  a0220a^3-  +  «0202:''4-) 

+  12  2/3-(rt202oa'i-  +  no22QXr'  +  «oo40»"3-  +  «oo22a:4-) 

+  12  2/4-(«2002A"l-  +  a0202a-2'-^  +  «0022X3^  +  «0004-^"4"'^) 

+  48  tjiyiarmXiXi  +  48  yxijsaim'XiXi  +  48  ?/i?/4«2002a^ia^4 
+  48  y^yiaxt'i-ioXiXz  +  48  yiyiUma-i^iXi  +  48  (/3.V4«oo22a:3^4- 
^uV(^)  =  24(?/i3a4oooXi  +  y2^«040oa:^2  +  2/3^«oo4oa;3  +  2/4''rtooo4a;4 
+  3  ?/i-*/2a220o3'2  +  3  yiy2-n22ooXi  +  3  yi'-2/3«202oa'3 

+  3  ?/l?/3-a2020-*'l  +  3  .'/l-l/4«2002-T4  +  3  2/l2/4-«2002a;i 

+  3  ?/2-'2/3ao22oa;3  +  3  y2yrao22oX2  +  3  !/22/y4ao202.*'4 
+  3  y22/4-«0202a^2  +  3  2/3-J/4«oo22a-4  +  3  2/32/4-aoo22a;3)  • 
A„V(a;)  =  24/(2/). 

Page  209.     Art.  164 

1.    (1,  0,  0,  1),   (1,  0,  0,  -  1),   (4,  0,  0,  -  1). 

2    4     2359  +  1 31  ■v/17      2359-  131  Vl7 
'  376  '  376 

Page  211.     Art.  167 

1.  (tOOOn  =  0)       aiWn-lXl  +  flolO  n-lX2   4"  «n01  h-1-^3  =  0- 

2.  osooon  =  0,     aioon-1  =  0)     rtoiOrt-i  =  0,     aonin-i  =  0. 

3.  2(3:1-3-3)  +5(a;2-X4)  =0. 

4.  2(a;i-a;3)+ 5(a;2-a-4)  =0,    4  xi  +  32  X2  -  36  3-4  ±  Vl042(a;2  -0:4)  =  0. 

Page  213.     Art.  169 

1.  (X2^  +  0:3^  +  X4^y-  =  0.  3       ,,^1  ^  „^i  +  ,,3^   ^  ^J  ^  0. 

2.  ?tl^  +  U2^  +  1(3^  +  Ui^  =  0.  4.     Mi-?(3  +  U2'^Ui  =  0. 

Page  215.     Art.   172 

3.  J- + -i- 4- A_  +  Jl_  + 1 =^0.         4.   4(n-2)3. 

aixi      a2X2      azxz      04X4      ai{x\  +  0^2  +  X3  +  X4) 

Page  218.     Art.   175 

2.  X\  =  0,  3:42  —  X2%z  =  0  ;  X2  —  X3  =  0,  X2  —  X4  =  0  ; 

X1X2  —  X1X3  +  X42  —  X2a;3  =  0,  xi^  —  X2a-4  —  X2'-^  +  X2X3  =  0, 
x^  +  a;2a;3  +  2  X2X4  +  X3X4  —  xiX2  =  0. 

3.  (Xl2  +  X22  -  X32)2  -  4(Xi  -  X2)  (Xi3  +  X-?  +  X^Xa^  -  2  X1X32)  =  0. 

4.  x^  H-  X22  +  5  X32  =  0. 

5.  (ai  —  a4)xr  +  («2  —  «4)x22  +  {a%  —  ai^X'^  —  0. 
7.    (xi2- 2  X22+X32  + 2x42 -2x2X3)2 

-  2(xi  _  3  X2  +  2  X4)  [(xi2  +  X42  -  .T0X3)  (2  n  +  4  X2  -  2  X3) 

-  (xi2  +  2  X22  -  X32)  (2  xi  -  X2  -  .T3  +  2  X4)  ]  =  0. 


284  ANSWERS 

Page  225.     Art.  180 

1.  .Tl  =  t(t-  -  1),  X2  =  f-  -  1,  a-3  =  (<■-  -  1  )2,  X4  =  t. 

2.  (4  TsXi  —  xiX2y^  —  HX2^  +  2  a:ia:3)(a-i'-  +  2  X2a-4)  =  0. 

3.  12(ui^  -  U3^){U2^  —  M4^)—  l2(UiU2  —  If3«4)- 
+  (Mi2  +  2  Mo^  -  Ms'^  -  M4^)2  =  0. 

36(Mo2  _  M32)  (M22  -  M42)  (mi2  4-  2  tiz^  -  W32  -  M42) 

+  18(?<i2  -  4  ?t22  +  2  W32  +  2  ?<42)  (M1M2  -  UiUty 
-(?<l2  +  2  ?{22  -  Uz^  -  Ui^)^  =  0. 

Page  234      Art.  184 
8.   m  =  3,  n  =  3,  r  =  4,  iy  =  0,  /i  =  1,  G  =  0,  g  =  1,  a  =  0,  p  =  0, 

V  =  0,  u  =  0,  X  -  0,  y  =  0,  p  =  0. 
6.   On  the  developable  of  the  given  curve. 

Page  241.     Art.   187 

1.  The  four  quadric  cones  on  vyhich  d  lies. 

2.  Eight.     Four  of  each  regulus. 

4.  16  stationary  planes. 

24  planes  tangent  to  d  at  each  of  two  stationaiy  points. 

96  planes  tangent  at  one  and  passing  through  two  other  stationary  points 

116  planes  through  four  distinct  stationary  points. 

5.  The  developable  surface  of  C4.     The  four  quartic  curves  in  which  the 
faces  of  the  self-polar  tetrahedron  intersect  the  developable  surface. 

Page  243.     Art.  188 

I.  (a)  m  =  4,  n  =  6,  r  =  6,  H  =  1,  h  =  2,   G  =  0,  g  =  6,  a  =  4,  ^3  =  0, 

V  =  0,  w  =  0,  X  =  Q,  !/  =  4,  p  =  0. 

(b)  m  =4,   u  =  4,  r  ^  5,   H=  0,    h-2,   G  =  0,  gr  =  2,   a  =  1,    /3  =  1, 

V  =  0,  oj  =  0,  .X  =  2,  y  =  2,  p  ~  0. 

(c)  m  =  4,  n  =  6,  r  =  6,  H=0,  h  =  3,   G  =  0,  g  -6,    a  =  4,    /3  =  0, 

V  =  0,  w  =  0,  X  =  (k  ?/  =  4,  p  =  0. 

(d)  m  =  4,  /I  =  5,  r  =  6,  H=  0,  ^  =  3,    6?  =  0,    ^  =  4,  a  =  2,  /3  =  0, 

V  =  l,  w  =  0,  x  =  5,  ?/  =  4,  p  =  0. 

(e)  m-i,  n  =  4,  r  =  6,  Zf=0,  A  =  3,   G  =  0,  g  =  S,   a  =  0,   /3  =  0, 

V  =  2,  w  =  0,  X  =  4,  ?/  =  4,  p  =  0. 

4.    —  1,  2,  ^.        9.   Four.        10.    Four.     Two  of  each  regulus. 

II.  Ui  =  «3  _  3  (2  _  2,    U2  =it(t+   1)2,    7/3  =  -  «3,   ?<4  =  «3(f  +  1)2. 
Ml  =1,    M2  =—  2  «,     ?/.3   =  2  f'',    ?<^  =—  t,i. 


ANSWERS  285 


Page  253.     Art.   195 

1.  x  =  a  cos— ^,  y  =  a  sin— ^,  z  =  — ^• 

a\/2  a\/2  V2 

1      .        s  1  si 

2.  Tangent sin  -,   — -  cos -,   — -• 

\/2  rtV'2      v'2  aV2      V2 

Principal  normal  —  cos — '- — ,  —  sin -^ — ,  0. 
aV2  a\/2 

n-  ,       1  .  S  1  „  S  1 

Bmormal  —  sin , cos ,    — z- 

V2        ay/2         V2        aV2      V2 

p  =  2ffl,  ff  =—  2  a. 
S.    B  =  i-/'l±i?i±l^y(36(l  +4  fi  +  9t*)  +  (486  f  +  bQl  t^  f  90  t^-6  ty)l 
4.    (a)  No  curve.  (6)  A  cubic  curve. 

Page  267.     Art.  207 

1.   2  r<  cos  V  X  +  2  u  sinv  y  =  z  +  u"^, 
2  u  cos  V        2u  sin  v  —  1 


Vl  +  4  u'     Vl  +  4  tfi      Vl  +  4 1{2 
2.   dM2  +  M-2dy2  _  0. 

4.     {U  +  \/m2  -I-  cfi  +  6-2)  (?,  +  Vv2  +  0(2  ^.  52)  _  c, 

u  +  Vm2  +  a'''  +  62  =  c,i(ji2  +  Vu2"+a2~+^). 


INDEX 


The  numbers  refer  to  pages. 


Absolute,  53 
Angle,  3 

between  two  lines,  3,  22 

between  two  planes,  22 
Apolar,  182 
Axis,  radical,  47 

of  revolution,  50 

major,  mean,  minor,  63 

Binormal,  248 

Bundle  of  planes,  31,  115 

of  quadrics,  167 

parallel,  31 
Burnside  and  Panton,  239 

Center,  76 

of  ellipsoid,  63 

radical,  59 
Characteristic,  160 
Class  of  a  curve,  225 

of  a  surface,  210 
Cone,  49 

asymptotic,  96 

minimum,  190 

projecting,  217 

quadric,  72 

tangent,  212 
Conjugate  points,  132 

planes,  132 

point  and  line,  165 
Contragredient,  119 
Coordinates,  1 

curvilinear,  255 

cylindrical,  10 

elliptic,  106 

homogeneous,  33 

hyperbolic,  139 

plane,  31 

polar,  10 

spherical,  11 

tetrahedral,  109 
Correspondence,  120 

involutorial,  172 


Cross  ratio,  121 
Curvature,  248 

mean,  263 

total,  263 
Curve,  46 

algebraic,  215 

asymptotic,  261 

minimum,  252 

parametric,  255 

space,  215,  245 
Cusp,  226 
Cyclide,  203 

binodal,  204 

Dupin,  204 

horn,  204 

nodal,  203 

ring,  204 

spindle,  204 
Cylinder,  49 

elliptic,  72 

hyperbolic,  72 

imaginary,  72 

parabolic,  72 

projecting,  47 

Direction,  3 
cosines,  5 

Discriminant,  78,  126 

Discriminating  cubic,  79 

Distance,  4,  7 

between  two  lines,  24 
between  a  point  and  line,  23 
from  a  plane  to  a  point,  17 

Double  point  of  a  curve,  226 
apparent,  221 
of  a  surface,  203,  210 

Duality,  113 

Ellipse,  cubical,  235 
Ellipsoid,  63 

imaginary,  68 
Equation  of  plane,  12 

of  point,  32 


287 


288 


INDEX 


Equations  of  a  line,  19 

parametric,  138 
Euler,  42 

Factors,  invariant,  149 
Field,  plane,  115 
Fine,  216 

Formulas,  Euler's,  42 
Frenet-Serret,  250 

Generator,  94 
Genus,  228 

Halphen,  216 
Harmonic,  122 
Hessian,  212 
Horopter,  235 
Hyperbola,  cubical,  234 
Hyperboloid  of  one  sheet,  65 
of  two  sheets,  67 

Image,  139 

Independent  planes,  36 
Indicatrix,  267 
Inflexion,  linear,  226 
Intercepts,  13 
Invariant 

points,  121 

relative,  127 

under  motion,  82 
Inversion,  quadratic,  201 
Involution,  122 
Isotropic  planes,  54 

Jacobian  of  a  net,  170 
of  a  web,  176 

Kummer,  180 
surface,  180 

Law  of  inertia,  136 
Lines,  conjugate,  134 

minimum,  190 

normal,  255 

of  centers,  76 

of  curvature,  263 

of  vertices,  76 

Matrix,  37 
Meusnier,  260 
Monoid,  219 

Node,  226 
Noether,  221 
Normal,  92 


Normal  form,  13 
principal,  247 

Octant,  2 

Order  of  curve,  170 

of  surface,  208 
Origin,  1 

Parabola,  cubical,  235 
Paraboloid,  elliptic,  69 

hyperbolic,  70 
Parameter,  21 
Parametric  equations,  21 
Pencil  of  planes,  26,  115 

of  quadrics,  147 
Perspectivity,  196 
Plane,  13 

at  infinity,  76 

diametral,  75 

double  osculating,  226 

fundamental,  73 

normal,  132,  208 

of  centers,  76 

principal,  78 

radical,  57 

rectifying,  247 

splf-conjugatc,  133 

stationary,  226 

tangent,  210 
Planes,  coordinate,  1 

isotropic,  190 

projecting,  26 
Point,  at  infinity,  21 

stationary,  226 
Points,  associated,  168 

circular,  53 

conjugate,  132,  153 

elliptic,  267 

fundamental,  197 

hyperbolic,  267 

imaginary,  44 

parabolic,  267 

self-conjugate,  133 
Polar  reciprocal  figures,  135 

tetrahedra,  135 
Position,  hyperbolic,  143 
Projection,  orthogonal,  3 

quadric  on  a  plane,  139 

quadric  cone  on  a  plane,  149 

stereographic,  59 

Quadric  cone,  72 
non-singular,  78 
singular,  78 
surface,  63,  124 


INDEX 


289 


Quadrics,  confocal,  104 
Quartic  curve,  235 

first  kind,  242 

non-singular,  238 
Quartic  curve,  rational,  240 

second  kind,  237 

Radii,  reciprocal,  201 
Radius  of  curvature,  249 

of  torsion,  249 
Range  of  points,  115 
Rank  of  curves,  224 

of  determinants,  37 

of  a  matrix,  37 
Reflection,  41 
Regulus,  94,  138 
Reye,  77 
Rotation,  38 

Salmon,  167,  177,  227 
Section,  circular,  98 
Semi-axis,  63 
Sphere,  52 

director,  93 

imaginary,  52 

osculating,  251 

point,  52 
Spheroid,  oblate,  64 

prolate,  65 
Steinerian,  214 
Surface,  46 

algebraic,  206 


Surface,  developable,  225 
of  revolution,  50 
polar,  208 
quadric,  63 

Tangent,  209 

double,  226 

inflexional,  226 

stationary,  226 
Tangents,  asymptotic,  261 

conjugate,  262 

inflexional,  210 
Tetrahedron,  coordinate,  35 

self-polar,  135 
Torsion,  248 
Transformation,  birational,  197 

of  coordinates,  38 

projective,  120 
Translation,  38 

Umbilic,  101 
Unit  plane,  110 
point,  110 

Vertex  of  bundle,  31 

of  quadric,  76 
Vertices  of  ellipsoid,  63 

Web,  176 
Weddle,  179 
surface,  179 


SCIENCE   AND   ENGINEERING 
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\jniversity  of  California, 
\               San   Diego 

\           DATE   DUE 

DEC  31 1977 

Aiir  f^T^QR? 

Auu  ♦**  ylavt. 

MAR  2  6  198^, 

MAY  1  2  1988  \ 

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SE  16 

UCSD  Libr. 

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